Primarily quasilocal fields and 1-dimensional abstract local class field theory
I.D. Chipchakov
Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer group Br$(L/E)$ equals the maximal subgroup of Br$(E)$ of exponent $p$, for every cyclic extension $L/E$ of degree $p$. The paper proves that finite abelian extensions of $E$ are uniquely determined by their norm groups and related essentially as in the classical local class field theory. This includes analogues to the fundamental correspondence, the local reciprocity law and the local Hasse symbol.
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\begin{document}

\title{Primarily quasilocal fields and 1-dimensional abstract local
class field theory}


\author
{I.D. Chipchakov}
\email{chipchak@math.bas.bg}
\address{Institute of Mathematics and Informatics\\Bulgarian Academy
of Sciences\\Acad. G. Bonchev Str., bl. 8\\1113, Sofia, Bulgaria}
\keywords{strictly primarily quasilocal field, field admitting
one-dimensional local class field theory, abelian extension, norm
group, Brauer group, cyclic algebra.
MSC 2000 Subject Classification: 12F10 (primary): 16K50; 12J10; 11S31}
\thanks{The author was partially supported by Grant MI-1503/2005 of the
Bulgarian NSF}
\begin{abstract}
Let $E$ be a field satisfying the following conditions: (i) the
$p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$
is a prime number for which $E$ is properly included in its maximal
$p$-extension; (ii) the relative Brauer group Br$(L/E)$ equals the maximal
subgroup of Br$(E)$ of exponent $p$, for every cyclic extension $L/E$ of
degree $p$. The paper proves that finite abelian
extensions of $E$ are uniquely determined by their norm groups and
related essentially as in the classical local class field theory.
This includes analogues to the fundamental correspondence, the local
reciprocity law and the local Hasse symbol.
\end{abstract}

\maketitle



\section{Introduction and statements of the main results}

This paper is concerned with finite abelian extensions of primarily
quasilocal (abbr, PQL) fields, and can be viewed as a continuation
of \cite{Ch6}, I. When $E$ is a strictly PQL-field, it shows that
these extensions and their norm groups are related as in the
fundamental correspondence of the classical local class field theory
(see Definition 1, \cite{I}, page 101, or Serre's chapter in
\cite{CF}). The paper proves that they are subject to an exact
analogue to the local reciprocity law (formulated, e.g., in
\cite{Sch}, Ch. 6, Theorem~8, and \cite{I}, Theorem~7.1), and to a
partial analogue to the local Hasse symbol (as characterized in
\cite{Ko2}, Ch. 2, see also \cite{Wh1}, and \cite{I}, Theorems~6.9
and 6.10) of form determined by invariants of the Brauer group
Br$(E)$. It takes a step towards characterizing the fields whose
finite abelian extensions have the above properties. When $E$
belongs to some special classes of traditional interest, the present
research enables one to achieve this aim and to find a fairly
complete description of the norm groups of arbitrary finite
separable extensions of $E$ (see Section 3, \cite{Ch5,Ch8} and the
references in \cite{Ch9}, Remark~3.9).

\medskip
The basic notation, terminology and conventions kept in this paper
are standard and virtually the same as in \cite{Ch6}, I, and \cite{Ch9}.
Throughout, every algebra $A$ is understood to be associative with a
unit lying in the considered subalgebras of $A$. Simple algebras are
supposed to be finite-dimensional over their centres, Brauer and
value groups are written additively, Galois groups are viewed as
profinite with respect to the Krull topology, and the considered
profinite group homomorphisms are continuous. For each field $E$, $E
^{\ast }$ denotes its multiplicative group, $E _{\rm sep}$ a
separable closure of $E$, $\mathcal{G} _{E} = \mathcal{G}(E _{\rm sep}/E)$
is the absolute Galois group of $E$, $s(E)$ stands for the class of
central simple $E$-algebras, $[B]$ denotes the similarity class of any $B
\in s(E)$, $d(E)$ is the subclass of division
algebras $D \in s(E)$, and $P(E)$ is the set of those prime numbers
$p$ for which $E (p) \neq E$, where $E (p)$ is the maximal
$p$-extension of $E$ in $E _{\rm sep}$. As usual, $E$ is said to be
formally real, if $-1$ is not presentable over $E$ as a finite sum
of squares; we say that $E$ is nonreal, otherwise. The field $E$ is
called Pythagorean, if it is formally real and the set $E ^{\ast 2}
= \{\l ^{2}\colon \ \l \in E ^{\ast }\}$ is additively
closed. For any field extension $F/E$, $\rho _{E/F}$ denotes the
scalar extension map of Br$(E)$ into Br$(F)$, and Br$(F/E)$ stands
for the relative Brauer group of $F/E$. When $F/E$ is finite and
separable, we write Cor$_{F/E}$ for the corestriction map of Br$(F)$
into Br$(E)$ (see \cite{T}). We say that $E$ is $p$-quasilocal, for a
given prime $p$, if it satisfies one of the following conditions:
(i) every cyclic extension of $E$ of degree $p$ is embeddable as a
subalgebra in each $\Delta \in d(E)$ of (Schur) index $p$; (ii) the
$p$-component Br$(E) _{p}$ of Br$(E)$ is trivial or $p \notin
P(E)$. The field $E$ is called PQL, if it is $p$-quasilocal for
every prime $p$; when this holds, $E$ is said to be strictly PQL in case
Br$(E) _{p} \neq \{0\}$, $p \in P(E)$. We say that $E$ is
quasilocal, if its finite extensions are PQL.

\medskip
It is well-known that local fields and $p$-adically closed fields
are strictly quasilocal (abbr, SQL), i.e. their finite extensions
are strictly PQL (cf. \cite{S2}, Ch. XIII, Sect. 3, \cite{PR}, Theorem~3.1 and
Lemma~2.9, and \cite{Ch4}, Sect. 3). The strictly PQL-property has
been fully characterized in the following two classes: (i)
algebraic extensions of global fields \cite{Ch8}; (ii) Henselian discrete
valued fields \cite{Ch4}, Sect. 2. These facts extend the arithmetic
basis of this research. However, it is motivated mainly by results
of \cite{Ch6}, I, on the Brauer groups of PQL-fields and on the absolute
Galois groups of quasilocal fields. For reasons clarified in the
sequel, our approach to the main topic of this paper is purely
algebraic. Our starting point is the fact that the residue fields of
Henselian valued stable fields are PQL in the case of totally
indivisible value groups (in the sense of (2.1) (i)). Specifically,
a Henselian discrete valued field $(K, v)$ with a perfect residue
field $\widehat K$ is stable if and only if $\widehat K$ is stable
and PQL (cf. \cite{Ch6}, I, and the references there). These and other
related results show, with their proofs, that PQL-fields partly
resemble local fields in a number of respects (see Propositions
2.2-2.3, \cite{Ch6}, I, Lemma~4.3 and \cite{Ch6}, II, Lemma~2.3). For
example, by \cite{Ch6}, I, Lemma~4.3, if $\widehat K$ is $p$-quasilocal and
$\widetilde L _{1}$ and $\widetilde L _{2}$ are different extensions
of $\widehat K$ in $\widehat K (p)$ of degree $p$, then the inner
product $N(\widetilde L _{1}/\widehat K)N(\widetilde L _{2}/\widehat
K)$ of the norm groups $N(\widetilde L _{i}/\widehat K)$, $i = 2$,
is equal to $\widehat K ^{\ast }$. This result and its key role in the
proof of \cite{Ch6}, I, Theorem~4.1, attract interest in the study
of the PQL-property along the lines of the classical local class
field theory, with the notion of a local field extended as follows:

\begin{defn}
Let $E$ be a field, Nr$(E)$ the set of norm groups of $E$, and
$\Omega (E)$ the set of finite abelian extensions of $E$ in $E _{\rm
sep}$. We say that $E$ admits 1-dimensional local class field theory
(abbr, LCFT), if the natural mapping of $\Omega (E)$ into Nr$(E)$
(by the rule $M \to N(M/E)$, $M \in \Omega (E)$)  is injective and
the following condition holds, for each $M _{1}$, $M _{2} \in \Omega
(E)$:

\smallskip
(1.1) The norm group (over $E$) of the compositum $M _{1}M _{2}$ equals the
intersection $N(M _{1}/E) \cap N(M _{2}/E)$, and $N(M _{1} \cap M
_{2}/E) = N(M _{1}/E)N(M _{2}/E)$.
\end{defn}

We say that $E$ is a field with 1-dimensional local $p$-class field
theory (abbr, local $p$-CFT), for a given prime $p$, if the fields
in the set $\Omega _{p} (E) = \{L \in \Omega (E)\colon \ L \subseteq
E(p)\}$ are uniquely determined by their norm groups and satisfy
condition (1.1). When this is the case and $p \in P(E)$, we have
Br$(E) _{p} \neq \{0\}$ (see Proposition 3.3). Observe also that $E$
admits LCFT if and only if it admits local $p$-CFT, for every $p \in
P(E)$. This follows from Lemma 2.1 and shows that PQL-fields with
LCFT are strictly PQL.

\smallskip
The main purpose of this paper is to shed light on the
place of strictly PQL-fields in LCFT by proving the following:

\begin{theo}
Strictly {\rm PQL}-fields admit {\rm LCFT}. Conversely, a field $E$
admitting such a theory is strictly {\rm PQL}, if each $D \in d(E)$ of
prime exponent $p$ is similar to a tensor product of cyclic division
$E$-algebras of index $p$.
\end{theo}

\begin{theo}
Let $E$ and $M$ be fields, such that $E$ is strictly {\rm PQL},
$P(E) \neq \phi $  and $M \in \Omega (E)$. For each $p \in P(E)$,
let $_{p} {\rm Br}(E)$ be the maximal subgroup of {\rm Br}$(E)$ of
exponent $p$, $I _{p}$ a basis and $d(p)$ the dimension of $_{p}
{\rm Br}(E)$ as a vector space over the field $\FF _{p}$ with $p$
elements, $\mathcal{G}(M/E) _{p}$ the Sylow $p$-subgroup of the
Galois group $\mathcal{G}(M/E)$, and $\mathcal{G}(M/E) _{p}
^{d(p)}$ the direct product, indexed by $I _{p}$, of isomorphic
copies of $\mathcal{G}(M/E) _{p}$. Then the direct product
$\mathcal{G}(M/E) ^{{\rm Br}(E)} = \prod _{p \in P(E)}
\mathcal{G}(M/E) _{p} ^{d(p)}$ and the quotient group $E ^{\ast
}/N(M/E)$ are isomorphic.
\end{theo}

Before stating our third main result, recall that a field $F$ is
Pythagorean with $F (2) = F(\sqrt{-1})$ if and only if it is
formally real and $2$-quasilocal \cite{Ch6}, I, Lemma~3.5. Note also
that, by \cite{Wh2}, Theorem~2, if $p \in P(F)$, then $F (p)$ contains as
a subfield a $\ZZ _{p}$-extension $U _{p}$ of $F$ (i.e. $U
_{p}/F$ is Galois with $\mathcal{G}(U _{p}/F)$ isomorphic to the
additive group $\ZZ _{p}$ of $p$-adic integers) unless $p
= 2$ and $F$ is Pythagorean. We retain notation as in Theorem 1.2.

\begin{theo}
Let $E$ be a strictly {\rm PQL}-field, such that $P(E) \neq \phi $,  and let
$E _{\infty } \subseteq E _{\rm sep}$ be the compositum of fields $E
_{p}$, $p \in P(E)$, where $E _{p}/E$ is a $\ZZ _{p}$-extension, if $p >
2$ or $E$ is nonreal, and $E _{2} = E (2)$ when $E$ is formally real.
Then there exists a set $H _{E} = \{( \ , M/E)\colon \ E ^{\ast } \to
\mathcal{G}(M/E) ^{{\rm Br}(E)}$, $M \in \Omega (E)\}$ of surjective
group homomorphisms with the following properties:

{\rm (i)} The kernel of $( \ , M/E)$ coincides with $N(M/E)$, for
each $M \in \Omega (E)$;

{\rm (ii)} If $M \in \Omega (E)$ and $K$ is an intermediate field of
$M/E$, then $( \ , K/E)$ equals the composition $\pi _{M/K} \circ (
\ , M/E)$, where $\pi _{M/K}\colon {\mathcal G}(M/E) ^{{\rm Br}(E)} \to
\mathcal{G}(K/E) ^{{\rm Br}(E)}$ is the homomorphism mapping the $i
_{p}$-th component of
\par\noindent
${\mathcal G}(M/E) _{p} ^{d(p)}$ on the $i
_{p}$-th component of $\mathcal{G}(K/E) _{p} ^{d(p)}$ as the natural projection
$\mathcal{G}(M/E) _{p} \to \mathcal{G}(K/E) _{p}$, for each pair $(p, i
_{p}) \in P(E) \times I _{p}$;
\par
{\rm (iii)} The set $H _{E}$ is uniquely determined by the mappings
$( \ , \Gamma /E)$, where $\Gamma $ runs through the set of finite
extensions of $E$ in $E _{\infty }$ of primary degrees.
\end{theo}

Theorems 1.1 and 1.2 show the strong influence of Br$(E)$ on a
number of algebraic properties of a PQL-field $E$. They are
obtained from similar results on finite abelian $p$-extensions of
$p$-quasilocal fields, stated as Theorem 3.1. This approach uses
several properties of $p$-quasilocal fields without generally
valid analogues for PQL-fields (see Propositions 2.2 and 2.3,
\cite{Ch6}, I, Corollary~8.5, and \cite{Ch9}, Proposition~6.3). As
shown in \cite{Ch9} and \cite{Ch6}, II, Sect. 3, it enables one to
describe the isomorphism classes of Brauer groups of major types of
PQL-fields, and of the reduced parts of Brauer groups of
equicharacteristic Henselian valued absolutely stable fields with
totally indivisible value groups. Thus it turns out that usually
powerful methods of valuation theory are virtually inapplicable to
many PQL and most presently known quasilocal fields (see (2.4) (iii)
and (2.3) with its proof, and compare (2.1) with (2.2) and Remark
2.4). At the same time, the proofs in \cite{Ch9} show at crucial
points that the study of the PQL-property can effectively rely on
constructive methods based on properties (established in the 1990's,
see e.g. \cite{FSS}, Proposition~2.6, and \cite{SVdB}) of relative
Brauer groups of extensions obtained as transfers of function fields
of Brauer-Severi and other varieties. This makes it possible to
apply Theorem 3.1 and other results about $p$-quasilocal fields to
Brauer groups of arbitrary fields, and so leads to a better
understanding of the relations between Galois groups and norm groups
of finite Galois extensions of quasilocal fields (see Remark 4.3 and \cite{Ch9}).

\smallskip
When $E$ is a local field, the former assertion of Theorem 1.1
yields the fundamental correspondence of the classical local class
field theory. If $E$ is merely strictly PQL with $d(p) =
1$, for all $p \in P(E)$ (i.e. with Br$(E)$ embeddable in
$\QQ /\ZZ $, the quotient group of the additive
group of rational numbers by the subgroup of integers), then Theorem
1.2 states that $E ^{\ast }/N(M/E) \cong \mathcal{G}(M/E)$. This holds,
for instance, in the following cases:

\medskip
(1.2) (i) $E$ is an algebraic strictly PQL-extension of a global
field $E _{0}$; when Br$(E) _{p} \neq \{0\}$, by \cite{Ch8},
Theorem~2.1, $p \in P(E)$ and Br$(E) _{p}$ is isomorphic to the
quasicyclic $p$-group $\ZZ (p ^{\infty })$ unless $p = 2$ and $E$ is
formally real (big families of such $E$ can be constructed by
applying \cite{Ch8}, Theorem~2.2).

(ii) The triple ${\rm AT}(E) = (\mathcal{G} _{E}, \{\mathcal{G} _{F}\colon
\ F \in \Sigma \}, E _{\rm sep} ^{\ast })$, $\Sigma $ being the set
of finite extensions of $E$ in $E _{\rm sep}$, is an Artin-Tate
class formation (see \cite{AT}, Ch. 14, and Remark 2.5 below).

\medskip
It is known that (1.2) (ii) holds, if $E$ is $p$-adically closed or
has a Henselian discrete valuation with a quasifinite residue field.
Note further that the statement of Theorem 1.2 coincides in case
(1.2) (ii) with the local reciprocity law of the Artin-Tate abstract
class field theory \cite{AT}, Ch. 14, Sect. 5. As to Theorem 1.3, it can be
viewed as a partial analogue to the local Hasse symbol (compare with
Theorem 3.2, Remark 7.1 and \cite{Wh1}, Theorem~3). The question of
whether fields with LCFT are strictly PQL is open. By Theorem 1.1,
its answer depends on the solution to one of the leading unsolved
problems in Brauer group theory (see Remark 3.4, \cite{MS}, Sect. 16,
and \cite{M}, Sect. 5). This allows us to prove in Section 3 that
SQL-fields are those whose finite extensions admit LCFT. Note also
that when $_{p} {\rm Br}(E)$ is finite, for each $p \in P(E)$, the
answer is positive if and only if $E ^{\ast }/N(M/E)$ $\cong {\mathcal
G}(M/E) ^{{\rm Br}(E)}$, $M \in \Omega (E)$ (apply Theorem 1.2 and
\cite{P}, Sect. 15.1, Proposition~b).

\medskip
The paper is organized as follows: Section 2 includes generalities
about PQL-fields. The former and the latter assertions of Theorem
1.1 are proved in Sections 4 and 3, respectively. Our main results
on local $p$-CFT are stated in Section 3. Theorems 1.2, 1.3 and
these results are proved in Sections 4, 6 and 7. Section 5 contains
an interpretation of a part of local $p$-CFT in terms of Galois
cohomology. It generalizes \cite{LLMS}, Theorem~1, as well as
known relations between local fields and Demushkin groups (cf.
\cite{S1}, Ch. II, Theorem~4), and the sufficiency part of the main
results of \cite{MW1,MW2}.



\section{Preliminaries on PQL-fields}

The present research is based on the possibility to reduce the
study of norm groups of finite abelian extensions to the
special case of $p$-extensions. The reduction is obtained by applying
the following lemma (which can be deduced from Galois theory (see
e.g., \cite{L}, Ch. VIII) and \cite{Ch6}, II, Lemma~2.2).


\begin{lemm}
Let $E$ be a field, $M \in \Omega (E)$, $M \neq E$, $\Pi $ the set
of prime divisors of $[M\colon E]$, and $M _{p} = M \cap E (p)$,
for each $p \in \Pi $. Then $M$ coincides with the compositum of the
fields $M _{p}\colon \ p \in \Pi $, $N(M/E) = \cap _{p \in \Pi } N(M
_{p}/E)$ and $E ^{\ast }/N(M/E)$ is isomorphic to the direct product
$\prod _{p \in \Pi } E ^{\ast }/N(M _{p}/E)$.
\end{lemm}

\medskip
The main results of \cite{Ch6}, I, used in this paper can be
stated as follows:

\medskip
\begin{prop} Let $E$ be a $p$-quasilocal field with {\rm Br}$(E) _{p}
\neq \{0\}$, for some $p \in P(E)$. Suppose further that $R$ is a
finite extension of $E$ in $E (p)$ and $D \in d(E)$ is an algebra of
$p$-primary dimension. Then:

{\rm (i)} $R$ is $p$-quasilocal and $D/E$ is a cyclic of exponent
{\rm exp}$(D)$ equal to {\rm ind}$(D)$;

{\rm (ii)} Br$(R) _{p}$ is a divisible group unless $p = 2$, $R = E$
and $E$ is formally real; when $E$ is formally real, $E (2) =
E(\sqrt{-1})$ and {\rm Br}$(E) _{2}$ is of order $2$;

{\rm (iii)} $\rho _{E/R}$ maps {\rm Br}$(E) _{p}$ surjectively on
{\rm Br}$(R) _{p}$ and {\rm Cor}$_{R/E}$ maps {\rm Br}$(R) _{p}$
injectively in {\rm Br}$(E) _{p}$; every $E$-automorphism $\psi $ of
the field $R$ is extendable to a ring automorphism on each $D _{R}
\in s(R)$ of $p$-primary index;

{\rm (iv)} $R$ embeds in $D$ as an $E$-subalgebra if and only
if the degree $[R\colon E]$ divides {\rm ind}$(D)$; $R$ is a
splitting field of $D$ if and only if ${\rm ind}(D) \mid [R\colon E]$.
\end{prop}

\medskip
Our next result gives an equivalent form of Proposition 2.2 (iv),
for PQL-fields, and shows its optimality in the class of algebraic
strictly PQL-extensions of the field $\QQ $ of rational numbers. We
refer the reader to \cite{Ch6}, II, for a proof of this result,
which demonstrates the applicability of the arithmetic method of
constructing such extensions, based on \cite{Ch8}, Theorem~2.2.

\medskip
\begin{prop}
{\rm (i)} Let $E$ be a {\rm PQL}-field, $M/E$ a finite
Galois extension and $R$ an intermediate field of $M/E$. If ${\mathcal
G}(M/E)$ is nilpotent, then $R$ embeds as an $E$-subalgebra in each
$D \in d(E)$ of index divisible by $[R\colon E]$.
\par
{\rm (ii)} For each nonnilpotent finite group $G$, there exists a
strictly {\rm PQL}-field $\Psi = \Psi (G)$, such that $E/\QQ $ is an
algebraic extension, {\rm Br}$(\Psi ) \cong \QQ /\ZZ $ and there is a
Galois extension $\Psi ^{\prime }$ of $\Psi $ with $\mathcal{G}(\Psi
^{\prime }/\Psi ) \cong G$, which does not embed as a $\Psi
$-subalgebra in any $\Delta \in d(\Psi )$ of index {\rm ind}$(\Delta )
= [\Psi ^{\prime }\colon \Psi ]$.
\end{prop}

The main results of \cite{Ch6}, I, and \cite{Ch2}, Sect. 3, show that
Br$(E)$ and $\mathcal{G} _{E}$ are the main algebraic structures
associated with any quasilocal field $E$. Therefore, we would like
to point out that $\mathcal{G} _{E}$ is prosolvable and Br$(E)$ is
embeddable in $\QQ/\ZZ$ in the following two cases:

\medskip
(2.1) (i) $E$ is an algebraic extension of a quasilocal field $K$
with a Henselian valuation $v$, such that $v(K)$ is totally
indivisible (i.e. $v(K) \neq pv(K)$, for every prime $p$) and finite
extensions of $K$ in $K _{\rm sep}$ are defectless; then Br$(E)$ is
divisible with Br$(E) _{p'} = \{0\}$, $p ^{\prime } \notin
P(E)$ (the prosolvability of $\mathcal{G} _{E}$ is implied by Galois
theory and \cite{Ch1}, Proposition~3.1, and it holds without the
assumption on the finite extensions of $K$ in $K _{\rm sep}$; the
assertion about Br$(E)$ can be deduced from \cite{Ch1},
Theorem~2.1, \cite{S2}, Ch. II, Proposition~6, the lifting property of
Henselian valuations, and \cite{Ch6}, I,  (1.3)).
\par
(ii) $E$ is formally real and quasilocal; then Br$(E)$ is of order
$2$ and $\mathcal{G} _{E}$ has an abelian open subgroup of index
$2$, which is either procyclic or $2$-generated as a topological group
(see \cite{Ch2}, Sect. 3, and \cite{Ch9}, Proposition~3.4).

\medskip
\noindent
Condition (2.1) (i) holds, for example, when $(K, v)$ is a Henselian
discrete valued quasilocal field or $E$ is an algebraic quasilocal
nonreal extension of a global field, such that Br$(E) \neq \{0\}$ (cf.
\cite{TY}, Proposition~2.2 and Theorem~3.1, and \cite{Ch8}, Sect. 3,
respectively). The existence of a PQL-field $F$ of essentially
nonarithmetic nature, i.e. with Br$(F)$ not embeddable in $\QQ /\ZZ $,
has been established in \cite{SVdB}, Sect. 3. As observed there, every
abelian torsion group $A$ embeds in Br$(F(A))$, for a suitably chosen
strictly PQL-field $F(A)$. This result and Proposition 2.2 (ii) are
complemented by the following statement which describes, in
conjunction with \cite{F}, Theorem 23.1, the isomorphism classes of
Brauer groups of some basic types of nonreal PQL-fields (see
\cite{Ch9}, Theorem~1.2 (i)-(ii), and for the formally real case, \cite{Ch9},
Proposition~6.4). It also shows that the absolute Galois groups of
nonreal SQL-fields need not be prosolvable and may have a very
complex structure:

\medskip
(2.2) An abelian torsion group $T$ is isomorphic to Br$(E)$, for
some nonreal PQL-field $E = E(T)$ if and only if $T$ is divisible.
If $T$ is divisible, $S$ is a set of finite groups, $E _{0}$ is an
arbitrary field, and $T _{0}$ is a subgroup of Br$(E _{0})$
embeddable in $T$, then $E$ can be chosen so as to satisfy
the following:

(i) $E$ is quasilocal with $\rho _{E/L}$ surjective, for any finite
extension $L/E$ (apply also the Albert-Hochschild-theorem (cf.
\cite{S1}, Ch. II, 2.2)).

(ii) $E/E _{0}$ is an extension, such that $E _{0}$ is algebraically
closed in $E$, $T _{0} \cap {\rm Br}(E/E _{0}) = \{0\}$ and every $G
\in S$ is realizable as a Galois group over $E$.

\medskip
Note that if $(F, v)$ is a Henselian valued field, such that $v(F)
\neq pv(F)$, for some prime $p$, then $v ^{\prime } (F
^{\prime }) \neq pv ^{\prime } (F ^{\prime })$ whenever $F
^{\prime }/F$ is a finite extension and $v ^{\prime }$ is a
valuation of $F ^{\prime }$ prolonging $v$. Let $G _{p}$ and
$\widehat G _{p}$ be Sylow pro-$p$-subgroups of $\mathcal{G} _{F}$
and $\mathcal{G} _{\widehat F}$, respectively, and let $F _{\rm in}$ be
the compositum of inertial finite extensions of $F$ in $F _{\rm
sep}$. Suppose further that finite extensions of $F$ are
$p$-quasilocal. Then, by the proof of \cite{Ch1}, Lemma~1.2,
$\widehat G _{p} = \{1\}$ or $\widehat G _{p} \cong \ZZ  _{p}$. In view
of \cite{La}, Theorem~3.16, and \cite{Ch2}, Lemma~1.1
(a), $\mathcal{G}(F (p)/F)$ is subject to the same alternative,
if $F$ is formally real and $p > 2$. At the same time, it follows from
\cite{Ch2}, Proposition 3.4, that $\widehat F$ and $F$ are nonreal in
case $p = 2$. Note that if $p \neq {\rm char}(\widehat F)$, then $G _{p}$
is of rank $r _{p} \leq 2$ (see \cite{Ch1}, Remark~1.4). When $p = {\rm
char}(\widehat F)$ and $p \in P(\widehat F)$, the proof of \cite{Ch1},
Lemma~1.3, shows that each $D ^{\prime } \in d(F ^{\prime })$ is
defectless over $F ^{\prime }$ with respect to $v ^{\prime }$. By \cite{TY},
Theorem~3.1, this implies defectlessness of finite extensions of $F$ in $F
_{\rm sep}$ (with respect to $v$). Observe also that $L \subseteq F _{\rm
in}$, if $L$ is a finite Galois extension of $F$ in $F _{\rm sep}$, such that
$\mathcal{G}(L/F)$ is nonabelian and simple. This follows from Galois
theory, the prosolvability of $\mathcal{G} _{F _{\rm in}}$ and the
normality of $F _{\rm in}/F$ (see \cite{JW}, page 135, and, e.g., \cite{Ch1},
page 3102). These results, the simplicity of the alternating
groups $A _{n}$, $n \geq 5$, and the embeddability of groups of
order $n$ in $A _{n+2}$, for each $n \in \NN $, enable
one to prove the following:

\medskip
(2.3) A field $E$ does not admit Henselian valuations with
indivisible value groups in the following two cases: (i) $E$ is
quasilocal and the groups $A _{n}$, $n \geq 5$, are isomorphic to
subquotients of $\mathcal{G} _{E}$ (by open normal subgroups); (ii) $E$
is formally real and PQL with $\mathcal{G}(E (p)/E)$ a pro-$p$-group of
rank $r(p) \geq 2$ whenever $p \geq 3$ is a prime number for which $G
_{E}$ is of nonzero cohomological $p$-dimension cd$_{p} (\mathcal{G}
_{E})$. In both cases, every $D \in d(E)$ is inertial with respect to any
Henselian valuation of $E$ (see also the proof of \cite{Ch9}, Corollary~6.7).

\begin{rema}
Let $T$ be a divisible abelian torsion group, $\overline P$ the set
of prime numbers and Supp$(T)$ the set of those $p \in \overline P$
for which the $p$-component $T _{p}$ of $T$ is nontrivial. Fix a
field $E$ with Br$(E) \cong T$ as in (2.2) (i) and denote by $E
_{\rm sol}$ the maximal Galois extension of $E$ in $E _{\rm sep}$
with a prosolvable Galois group. Then $E _{\rm sol}/E$ possesses an
intermediate field $E ^{\prime }$ that is strictly PQL with Br$(E
^{\prime }) \cong T$ (take as $E ^{\prime }$, e.g., the fixed field
of some Hall pro-${\rm Supp}(T)$-subgroup of $\mathcal{G}(E _{\rm
sol}/E)$). Note also that if Supp$(T) = \overline P$, then $E$ is
SQL.
\end{rema}

Statement (2.3) (i) is in fact a partial conversion of the
Koenigsmann-Neukirch theorem (cf. \cite{Koe}, Theorem~B).
Altogether, (1.2) (i), (2.1)$\div $(2.3) and Remark 2.4 suggest usually a
limited role of arithmetic, orderings and Henselian valuations in the
study of SQL-fields. At the same time, they draw interest in the following
open questions:

\medskip
(2.4) (i) Describe the isomorphism classes of Brauer groups of nonreal
SQL-fields.

(ii) Let $E$ be an SQL-field such that $\mathcal{G} _{E}$ is prosolvable.
Find whether the nonzero components Br$(E) _{p}$ of Br$(E)$, except possibly
one,  are isomorphic to $\ZZ (p ^{\infty })$. Also, find whether the Sylow
pro-$p$-subgroups of $\mathcal{G} _{E}$ are of rank $r _{p} \le 2$, for all prime
numbers $p$, with at most $2$ exceptions.

(iii) Let $F$ be a quasilocal field with Br$(F) _{\pi }$ and Br$(E)
_{\pi '}$ not embeddable in $\QQ/\ZZ $, for some different prime
numbers $\pi $ and $\pi ^{\prime }$. Find whether $F$ has a Henselian valuation
$v$ such that $v(F)$ is indivisible.

\medskip
\noindent
Questions (2.4) (ii) are closely related by Galois cohomology (cf.
\cite{MS}, (11.5), \cite{W1}, Theorem 3.1, and \cite{W2}, Lemma 7). Note
also that, by the comment at the end of \cite{Ch1}, Sect. 1, the
assumptions of (2.1) (i) guarantee that  $r _{p} \le 2$, for every prime $p
\neq {\rm char}(\widehat E)$ (here the condition on the finite extensions of
$K$ in $K _{\rm sep}$ is superfluous). In view of \cite{Koe}, Theorem
B, it is likely that a negative answer to the former question in (2.4) (ii) would
require the existence of a field $F$ with Br$(F) \neq \{0\}$ whose nontrivial
valuations have Henselizations equal to $F _{\rm sep}$ (which would solve
affirmatively Problem 11.5.9 (b) in \cite{FJ}).

\smallskip
\begin{rema}
It is known (see \cite{NP}) that if $E$ is a field, then the triple AT$(E)$
defined in (1.2) (ii) is an Artin-Tate class formation if and only if $E$ is
SQL, Br$(E)$ is embeddable in $\QQ/\ZZ$, and $\rho _{E/R}$ is
surjective, for every finite extension $R$ of $E$ in $E _{\rm sep}$.
Observing that (2.2) provides access to the richest presently known
sources of such fields, we add to the examples given after the
statement of (1.2) that AT$(E)$ satisfies the noted condition when
$E$ is an algebraic SQL-extension of a global field (see \cite{NP} and
\cite{Ch8}, Sect. 3).
\end{rema}



\section{Statements of the main results of local $p$-class field theory}

Our main results on local $p$-CFT are stated as the following
two theorems:

\smallskip
\begin{theo} Let $E$ be a $p$-quasilocal field with {\rm Br}$(E)
_{p} \neq \{0\}$, for some $p \in P(E)$. Suppose that  $\Omega
_{p} (E)$ is defined as in Section 1, and let $I _{p}$ and $d(p)$ be as
in Theorem 1.2. Then $E$ admits local $p$-{\rm CFT}, and for each $M
\in \Omega _{p} (E)$, $E ^{\ast }/N(M/E)$ is presentable as a direct
product $\mathcal{G}(M/E) ^{d(p)}$ of isomorphic copies of ${\mathcal
G}(M/E)$, indexed by $I _{p}$.
\end{theo}


Theorem 3.1 and Lemma 2.1 imply Theorem 1.2 and the former assertion
of Theorem 1.1. Note that Theorem 3.1 contains exact analogues to
the fundamental correspondence and the local reciprocity law.
Similarly, our next result can be considered as such an analogue to
Hasse's symbol for local fields (see \cite{Ko2}, Ch. 2, as well as
Corollary 7.3 and Remark 7.1).


\begin{theo} Under the hypotheses of Theorem 3.1, let $E _{\infty }$
be a $\ZZ _{p}$-extension of $E$ in $E (p)$, and for any
$n \in \NN $, let $\Gamma _{n}$ be the intermediate field
of $E _{\infty }/E$ of degree $[\Gamma _{n}\colon E] = p ^{n}$. Then there exist
sets $H _{p} (E ^{\prime }) = \{( \ , M ^{\prime }/E ^{\prime
})\colon \ E ^{\prime \ast } \to \mathcal{G}(M ^{\prime }/E ^{\prime })
^{d(p)}, M ^{\prime } \in \Omega _{p} (E ^{\prime })\}$,  $E
^{\prime } \in \Omega _{p} (E)$, of surjective group homomorphisms
satisfying the following:

{\rm (i)} The kernel of $( \ , M ^{\prime }/E ^{\prime })$ is equal
to $N(M ^{\prime }/E ^{\prime })$, for each $E ^{\prime } \in
\Omega _{p} (E)$, $M ^{\prime } \in \Omega _{p} (E ^{\prime })$;

{\rm (ii)} If $M \in \Omega _{p} (E)$ and $K$ is an intermediate
field of $M/E$, then $( \ , K/E) = \pi _{M/K} \circ ( \ , M/E)$, where $\pi
_{M/K}\colon \ \mathcal{G}(M/E) ^{d(p)} \to \mathcal{G}(K/E) ^{d(p)}$
is the homomorphism acting componentwise as the natural
mapping of $\mathcal{G}(M/E)$ on $\mathcal{G}(K/E)$;

{\rm (iii)} In the setting of (ii), $(\l ,
M/K) = (N _{E} ^{K} (\l ), M/E)$, for each $\l \in K
^{\ast }$;

{\rm (iv)} The sets $H _{p} (E ^{\prime }), E ^{\prime } \in \Omega
_{p} (E)$, are determined by the sequence

\noindent
$( \ , \Gamma _{n}/E)\colon \ n \in \NN $.
\end{theo}

\medskip
The rest of this Section concerns the question of whether fields
with LCFT are strictly PQL. Lemma  2.1 and our first result on this
question imply the latter assertion of Theorem 1.1.

\begin{prop}
Let $E$ be a field admitting local $p$-{\rm CFT}, for some $p \in
P(E)$, and let $L$ be an extension of $E$ in $E (p)$ of degree $p$.
Then {\rm Br}$(L/E)$ is nontrivial and independent of the choice of
$L$. Furthermore, if each $D \in d(E)$ of exponent $p$ is similar
to tensor products of cyclic division $E$-algebras of index $p$,
then $E$ is $p$-quasilocal.
\end{prop}

\begin{proof}
Our assumptions ensure that $L/E$ is cyclic, whence
Br$(L/E) \cong E ^{\ast }/N(L/E)$ (cf. \cite{L}, Ch. I, Sect. 6, and
\cite{P}, Sect. 15.1, Proposition~b). As $E$ admits local $p$-CFT
and $L \neq E$, this yields $N(L/E) \neq N(E/E) = E ^{\ast }$ and
Br$(L/E) \neq \{0\}$. In order to complete our proof, it suffices to
show that $L$ embeds as an $E$-subalgebra in each cyclic division
$E$-algebra of index $p$. If $\mathcal{G}(E (p)/E)$ is procyclic, this
is evident, so we assume that $\Omega _{p} (E)$ contains a field $F
\neq L$, such that $[F\colon E] = p$. It follows from Galois theory that
$LF \in \Omega (E)$, $[LF\colon E] = p ^{2}$ and $\mathcal{G}(LF/E)$ is
noncyclic. Therefore, $LF/E$ possesses $p + 1$ intermediate fields
of degree $p$ over $E$. Let $F ^{\prime }$ be such a field different
from $L$ and $F$. Then $N(F/E)N(F ^{\prime }/E) = E ^{\ast }$ and
$LF ^{\prime } = LF$. Moreover, the norm maps $N _{E} ^{F}$ and
$N _{E} ^{F'}$ are induced by $N _{L} ^{LF}$, so it turns out that $E
^{\ast } \subseteq N(LF/L)$. Hence, by \cite{P}, Sect. 15.1,
Proposition~b, $L$ embeds over $E$ in each $\Delta \in d(E)$ split
by $F$, which proves Proposition 3.3.
\end{proof}

\begin{rema}
Let $E$ be a field and $p \in P(E)$.
\par
(i) If $E (p)/E$ is a $\ZZ _{p}$-extension and $L$ is the
unique subextension of $E$ in $E (p)$ of degree $p$, then $E$
admits local $p$-CFT if and only if Br$(L/E) \neq \{0\}$ (apply
Proposition 3.3 and \cite{P}, Sect. 15.1, Corollary~b).
\par
(ii) When Br$(E) _{p} \neq \{0\}$, the concluding condition of
Proposition 3.3 is satisfied in the following cases: (a) $E$ is an
algebraic extension of a global or local field $E _{0}$; (b) $E$
contains a primitive $p$-th root of unity or char$(E) = p$; (c) $p =
3, 5$. In cases (b) and (c), this follows from the Merkurjev-Suslin
theorem \cite{MS}, (16.1), \cite{A}, Ch. VII, Theorem 28, \cite{M},
Sect. 4, Corollary, and \cite{Ma}. In case (a), by class field
theory, every $A _{0} \in s(E _{0})$ is cyclic (cf. \cite{AT}, Ch.
10, Theorem~5), which implies the same, for all $A \in s(E)$.
\par
(iii) It is not known whether $E$ is strictly PQL, if it admits LCFT
and has a Henselian discrete valuation (see \cite{Ch4}, Sect. 2).
\end{rema}

\begin{coro}
A finite purely inseparable extension $K$ of a field $E$ of
characteristic $p > 0$ admits local $p$-{\rm CFT} if and only if $E$ has
the same property.
\end{coro}

\begin{proof}
By \cite{Ch6}, I, Proposition~4.4, $E$ is $p$-quasilocal if and only
if $K$ is $p$-quasilocal, and by the Albert-Hochschild theorem, $\rho
_{E/K}$ is surjective. Since the exponent of Br$(K/E)$ divides
$[K\colon E]$ (see \cite{P}, Sects. 13.4 and 14.4), and by Witt's
theorem (cf. \cite{Dr}, page 110), Br$(E) _{p}$ and Br$(K) _{p}$ are
divisible, this implies that Br$(E) _{p} \neq \{0\} \leftrightarrow {\rm
Br}(K) _{p} \neq \{0\}$, so Corollary 3.5 follows from Theorem 3.1 and
Remark 3.4 (ii).
\end{proof}

The concluding result of this Section clarifies the role of
SQL-fields in LCFT. In view of (2.2), Remark 2.5 and the main results
of \cite{Ch9}, it also determines the place of the Neukirch-Perlis
variant of the theory \cite{NP}, built upon (1.2) (ii), in the study of
strictly PQL-fields.

\begin{prop} A field $E$ is {\rm SQL} if and only if its finite
extensions admit {\rm LCFT}. When occurs, {\rm Br}$(E _{p}) \neq
\{0\}$, provided that $E _{p}$ is the fixed field of a Sylow
pro-$p$-subgroup of $\mathcal{G}_{E}$ and $p$ is a prime number for
which {\rm cd}$_{p} (\mathcal{G} _{E}) \neq 0$.
\end{prop}

\begin{proof}
The left-to-right implication is contained in Theorem 1.1, so we
prove the converse one and the nontriviality of Br$(E _{p}) _{p}$,
for a prime $p$ satisfying cd$_{p} (\mathcal{G} _{E}) \neq 0$.
Suppose that every finite extension $L$ of $E$ admits LCFT, and $B
_{p}$ is the extension of $E$ generated by the $p$-th roots of unity
in $E _{\rm sep}$. It is known (cf. \cite{L}, Ch. VIII, Sect. 3)
that $B _{p} \subseteq E _{p}$, i.e. $E _{p}$ contains a primitive
$p$-th root of unity unless $p = {\rm char}(E)$. Also, Proposition
3.3 and Remark 3.4 (ii) indicate that $L$ is $p$-quasilocal,
provided that $B _{p} \subseteq L$. As $p \dagger [L ^{\prime
}\colon E]$, for any finite extension $L ^{\prime }$ of $E$ in $E
_{p}$, this enables one to obtain from general properties of scalar
extension maps and Schur indices (cf. \cite{P}, Sect. 13.4, and
\cite{Ch6}, I, (1.3)) that $E _{p}$ is $p$-quasilocal. Moreover, it
follows from \cite{Ch6}, I, Lemma~8.3, and the choice of $p$ that
$E$ is quasilocal. Since, by Proposition 3.3, PQL-fields with LCFT
are strictly PQL, the obtained result shows that $E$ is SQL. Note
further that the inequality cd$_{p} (\mathcal{G} _{E}) \neq 0$
ensures the existence of finite Galois extensions of $E$ in $E _{\rm
sep}$ of degrees divisible by $p$. Therefore, by Sylow's theorems
and Galois theory, there is a finite extension $ L _{0}$ of $E$ in
$E _{p}$, such that $p \in P(L _{0})$. As in the proof of the
statement that $E _{p}$ is $p$-quasilocal, these observations imply
$p \in P(E _{p})$ and Br$(E _{p}) \neq \{0\}$.
\end{proof}

Let us note that the abstract approach to LCFT dates back to the early
1950's (cf. \cite{AT}, Sect. 14). It fits the character of the related
class formation theory, see \cite{K}, and accounts for the fact that the
Neukirch-Perlis variant of LCFT goes substantially beyond the
limits of (2.1). The fact itself is established by summing up Proposition
3.6, Remark 2.5, statements (2.1)$\div $(2.3), and the results on (2.4)
(ii) mentioned in Section 2.



\section{Proof of Theorem 3.1}

Let $E$ be a $p$-quasilocal field, for a given $p \in P(E)$. By
Proposition 2.2 (iv), then Br$(M/E) = \{b \in {\rm Br(E)}\colon \
[M\colon E]b = 0\}$, for every cyclic $p$-extension $M/E$. Hence, by
the structure of divisible abelian torsion groups, and by the fact that
Br$(M/E) \cong E ^{\ast }/N(M/E)$ (cf. \cite{F}, Theorem~23.1, and
\cite{P}, Sect. 15.1, Proposition~b), $E ^{\ast }/N(M/E) \cong {\mathcal
G}(M/E) ^{d(p)}$. To prove the obtained isomorphism, for any $M \in
\Omega _{p} (E)$, we need the following lemmas.

\begin{lemm}
Let $E$ be a $p$-quasilocal field, for some $p \in P(E)$, and let $E
_{1}, \dots E _{t}$ be cyclic extensions of $E$ in $E (p)$, for a given
integer $t \geq 2$. Assume that the compositum $E ^{\prime }$ of the
fields $E _{j}\colon \ j = 1, \dots , t$, satisfies the equality $[E ^{\prime
}\colon E] = \prod _{j=1} ^{t} [E _{j}\colon E]$. Then $N(E ^{\prime }/E)
= \bigcap _{j=1} ^{t} N(E _{j}/E)$.
\end{lemm}

\begin{proof}
The inclusion $N(E ^{\prime }/E) \subseteq \bigcap _{j=1} ^{t} N(E
_{j}/E)$ follows from the transitivity of norm maps in towers of
finite extensions (cf. \cite{L}, Ch. VIII, Sect. 5). Conversely, let $c \in
\bigcap _{j=1} ^{t} N(E _{j}/E)$, and $\b \in E
_{1} ^{\ast }$ be of norm $N _{E} ^{E _{1}} (\b ) = c$. The
equality $[E ^{\prime }\colon E] = \prod _{j=1} ^{t} [E _{j}\colon
E]$, Galois theory and \cite{Ch3}, Lemma~4.2, imply that $[E ^{\prime
}\colon E _{1}]  = \prod _{i=2} ^{t} [(E _{1}E _{i})\colon E _{1}]$
and $\b \in \bigcap _{i=2} ^{t} N(E _{1}E _{i}/E _{1})$. This
proves Lemma 4.1 in the case of $t = 2$. Since $(E _{1}E _{i})/E
_{1}$ is cyclic, for each $i \geq 2$, and by Proposition 2.2 (i), $E
_{1}$ is $p$-quasilocal, the obtained result makes it easy to
complete the proof of the lemma by induction on $t$.
\end{proof}

\begin{lemm}
Assume that $E$ is a $p$-quasilocal field, $L \in \Omega _{p} (E)$,
and $F$ is a cyclic extension of $E$ in $L$. Then $N(L/F)$ contains
the element $\psi (\a )\a ^{-1}$, for each $\a \in F ^{\ast }$ and any
$\psi \in \mathcal{G}(F/E)$.
\end{lemm}

\begin{proof}
As $F$ is $p$-quasilocal and $L \in \Omega _{p} (F)$, whence
${\mathcal G}(L/F)$ decomposes into a direct product of cyclic groups,
Galois theory and Lemma 4.1 allow one to consider only the case in
which $L/F$ is cyclic. Fix a generator $\s $ of $\mathcal{G}(L/F)$,
denote by $A _{\a }$ the cyclic $F$-algebra $(L/F, \s , \a )$, for an
arbitrary $\a \in F ^{\ast }$, put $m = [L\colon F]$, and take an invertible
element $\eta \in A _{\a }$ so that $\eta ^{m} = \a $ and $\eta \l \eta ^{-1}
= \s (\l )$, for every $\l \in L$. Also, let $\psi ^{\prime }$ be an
automorphism of $L$ extending $\psi $. Proposition 2.2 (iii) and the
Skolem-Noether theorem (cf. \cite{P}, Sect. 12.6) imply that $A _{\a }$
has a ring automorphism $\tilde \psi $, such that $\tilde \psi (\l )$ $= \psi
^{\prime } (\l )$, for any $\l \in L$. Since $\s \psi ^{\prime } = \psi ^{\prime
}\s $, this ensures that the element $\eta ^{-1}\tilde \psi (\eta ) := \mu $
lies in the centralizer of $L$ in $A _{\a }$. Thereby, we have $\mu
\in L ^{\ast }$ and $\tilde \psi (\eta ) ^{m} = \psi (\a ) =
\eta ^{m}N _{F} ^{L} (\mu ) = \a N _{F} ^{L} (\mu )$, which
proves Lemma 4.2.
\end{proof}

Assuming that $E$ is a $p$-quasilocal field, $M _{u} \in \Omega
_{p} (E)$, $[M _{u}\colon E] = p ^{\mu _{u}}\colon \ u = 1, 2$, and
putting $M ^{\prime } =  M _{1}M _{2}$, we prove the following
assertions:

\medskip
(4.1) (i) If $N(M _{1}/E) = N(M _{2}/E)$ and $M _{1} \subseteq M
_{2}$, then $M _{1} = M _{2}$;

(ii) If $M _{1} \cap M _{2} = E$, then $N(M _{1}/E) \cap N(M
_{2}/E) = N(M ^{\prime }/E)$,

\noindent
$N(M _{1}/E)N(M _{2}/E) = E ^{\ast }$,
and for each $\s \in \mathcal{G}(M _{2}/E)$, there is a unique $M
_{1}$-automorphism $\s ^{\prime }$ of $M ^{\prime }$ extending
$\s $; the mapping of $\mathcal{G}(M _{2}/E)$ on $\mathcal{G}(M
^{\prime }/M _{1})$ by the rule $\s \to \s ^{\prime }$ is an
isomorphism;
\par
(iii) Under the hypotheses of (ii), if $M _{2}/E$ is cyclic and $\mathcal{G}(M
_{2}/E) = \langle \s \rangle $, then $\mathcal{G}(M ^{\prime }/M _{1}) =
\langle \s ^{\prime } \rangle $, and Cor$_{M _{1}/E}$ maps Br$(M
^{\prime }/M _{1})$ into Br$(M _{2}/E)$ by the formula
$[(M ^{\prime }/M _{1}, \s ^{\prime }, \th )] \to [(M
_{2}/E, \s , N _{E} ^{M _{1}} (\th ))]$, $\th \in M _{1}
^{\ast }$; furthermore, if Br$(E) _{p}$ is infinite, then Cor$_{M
_{1}/E}$ induces isomorphisms Br$(M _{1}) _{p} \cong {\rm Br}(E)
_{p}$ and Br$(M ^{\prime }/M _{1}) \cong {\rm Br}(M _{2}/E)$.

\smallskip
It is sufficient to consider the special case where $M _{u}
\neq E\colon \ u = 1, 2$, and $M _{1} \neq M _{2}$. In view of
Proposition 2.2 (ii), then Br$(E) _{p}$ is divisible. Also, Galois theory and
the assumptions on $M _{1}/E$ imply the existence of a cyclic extension
$M _{0}$ of $E$ in $M _{1}$ of degree $p$. Suppose that $M _{1}
\subseteq M _{2}$ and $d \in N(M _{u}/E)\colon \ u = 1, 2$, fix a
generator $\psi $ of $\mathcal{G}(M _{0}/E)$ and elements $\eta _{j} \in M _{j}
^{\ast }$, $j = 1, 2$, so that $N _{E} ^{M _{1}} (\eta _{1}) = N _{E} ^{M
_{2}} (\eta _{2}) = d$. It follows from Hilbert's Theorem 90 that $N _{M
_{0}} ^{M _{1}} (\eta _{1}) = N _{M _{0}} ^{M _{2}} (\eta _{2})\psi
(\b )\b ^{-1}$, for some $\b \in M _{0} ^{\ast }$. Hence, by
Lemma 4.2, $N(M _{1}/E) = N(M _{2}/E)$ if and only if $N(M _{1}/M _{0}) =
N(M _{2}/M _{0})$. Observing also that $M _{0}$ is $p$-quasilocal and $[M
_{1}\colon M _{0}] = p ^{\mu _{1}-1}$, and proceeding by induction on
$\mu _{1}$, one proves (4.1) (i).

\par
Assume now that $M _{1} \cap M _{2} = E$. Then the Galois-theoretic
parts of (4.1) (ii) and (iii) are contained in \cite{L}, Ch. VIII,
Theorem~4. The rest of the former assertion of (4.1) (iii) is
implied by Proposition 2.2 (iii), the basic
restriction-corestriction (abbr, RC) formula (cf. \cite{T}, Theorem~2.5)
and the lemmas in \cite{Ch3}, Sect. 4. The equality $N(M
^{\prime }/E) = N(M _{1}/E) \cap N(M _{2}/E)$ follows from the
presentability of $M _{1}$ and $M _{2}$ as compositums of cyclic
extensions of $E$ satisfying the conditions of Lemma 4.1. It remains
for us to show that $N(M _{1}/E)N(M _{2}/E) = E ^{\ast }$ and to prove
the latter part of (4.1) (iii). Suppose first that $M _{2}/E$ is cyclic and the
automorphisms $\s , \s ^{\prime }$ are determined as in (4.1) (iii), fix an
element $c \in E ^{\ast }$ out of $N(M _{2}/E)$, and put $A _{c} =  (M
_{2}/E, \s , c)$. It is clear from \cite{P}, Sect. 15.1, Proposition~b, and the
choice of $c$ that ind$(A _{c}) \mid p ^{\mu _{2}}$ and ind$(A _{c}) > 1$. As Br$(E) _{p}$ is divisible, there exists
$\Delta _{c} \in d(E)$, such that $p ^{\mu _{1}}[\Delta _{c}] = [A _{c}]$.
In addition, it follows from Proposition 2.2 (i) that ind$(\Delta _{c}) = p
^{\mu _{1}}. {\rm ind}(A _{c})$. Observing also that $[M ^{\prime }\colon
E] = p ^{\mu }$ and ind$(\Delta _{c}) \mid p ^{\mu }$, where $\mu = \mu
_{1} + \mu _{2}$, one obtains from Proposition 2.2 (iv) that $M ^{\prime
}$ is a splitting field of $\Delta _{c}$. Since $\Delta _{c} \otimes
_{E} M ^{\prime } \cong (\Delta _{c} \otimes _{E} M _{1}) \otimes
_{M _{1}} M ^{\prime }$ as $M ^{\prime }$-algebras (cf. \cite{P},
Sect. 9.4, Corollary~a), this means that $[\Delta _{c} \otimes _{E} M
_{1}] \in {\rm Br}(M ^{\prime }/M _{1})$, or equivalently, that
$[\Delta _{c} \otimes _{E} M _{1}] = [(M ^{\prime }/M _{1}, \s
^{\prime }, \a )]$, for some $\a \in M _{1} ^{\ast }$.
Therefore, by Proposition 2.2 (iii), the RC-formula and the former
statement of (4.1) (iii), $[A _{c}] = [(M _{2}/E, \s , N _{E} ^{M
_{1}} (\a ))]$. As we have $[A _{c}\colon E] = [(M _{2}/E, \s , N
_{E} ^{M _{1}} (\a ))\colon E]$ $= [M _{2}\colon E] ^{2}$, this
proves that $A _{c} \cong (M _{2}/E, \s , N _{E} ^{M _{1}} (\a ))$
over $E$. Hence, by \cite{P}, Sect. 15.1, Proposition~b, $c.N _{E}
^{M _{1}} (\a ) ^{-1} \in N(M _{2}/E)$, which yields $N(M _{1}/E)N(M
_{2}/E) = E  ^{\ast }$ in case $M _{2}/E$ is cyclic. Our argument,
combined with Proposition 2.2 (ii)-(iii) and the RC-formula, also
proves the latter part of (4.1) (iii). Henceforth, we assume that $M
_{2}/E$ is noncyclic, i.e. $\mathcal{G}(M _{2}/E)$ is an abelian
$p$-group of rank $r \geq 2$. Then it follows from Galois theory and
the structure of finite abelian groups that there exist cyclic
extensions $F _{1}$ and $F _{2}$ of $E$ in $M _{2}$ such that $F
_{1} \cap F _{2} = E$ and the $p$-groups $\mathcal{G}(M _{2}/F
_{1})$ and $\mathcal{G}(M _{2}/F _{2})$ are of rank $r - 1$. As $M
_{1} \cap M _{2} = E$, one also sees that $(M _{1}F _{u}) \cap M
_{2} = F _{u}\colon \ u = 1, 2$. Taking now into account that $F
_{1}$ and $F _{2}$ are $p$-quasilocal fields, and arguing by
induction on $r$, one concludes that it suffices to deduce the
equality $N(M _{1}/E)N(M _{2}/E) = E ^{\ast }$ under the extra
hypothesis that $N(M _{1}F _{u}/F _{u})N(M _{2}/F _{u}) = F _{u}
^{\ast }\colon \ u = 1, 2$. Then, by norm transitivity in towers of
finite extensions, $N(F _{1}/E)N(F _{2}/E) \subseteq N(M _{1}/E)N(M
_{2}/E)$, and since $N(F _{1}/E)N(F _{2}/E) = E ^{\ast }$, this
completes the proof of (4.1).

\medskip
\begin{rema} Assume that $E$ is a field and $M
_{1}$, $M _{2}$ are finite extensions of $E$ in $E _{\rm sep}$, such
that $M _{2}/E$ is cyclic and $M _{1} \cap M _{2} = E$. It is known
that then the former assertion of (4.1) (iii) remains valid. The
cyclicity of $M ^{\prime }/M _{1}$ and the relations between ${\mathcal
G}(M _{2}/E)$ and $\mathcal{G}(M ^{\prime }/M _{1})$ follow from
\cite{L}, Ch. VIII, Theorem~4, and the validity of the formula about
the action of Cor$_{M _{1}/E}$ on Br$(M ^{\prime }/M _{1})$ can be
proved by using a group-cohomological technique (cf. \cite{We},
Proposition~4.3.7). It is worth noting, however, that (4.1) (iii)
plays a role in the proof not only of Theorems 3.1 and 3.2 but also
of Proposition 2.2 (i) (see \cite{Ch6}, I, Sect. 7). This, combined
with \cite{Ch9}, Theorem~1.2 (i)-(ii), enables one to find
alternative field-theoretic proofs of the formula in (4.1) (iii),
and of other results concerning Cor$_{F/E}$, for an arbitrary finite
separable extension $F/E$ (by reduction to the case where $E$
satisfies (2.2) (i), see \cite{Ch7}).
\end{rema}

\smallskip
We are now in a position to prove Theorem 3.1 (and thereby,
Theorem 1.2 and the former part of Theorem 1.1 as well). Assuming as
above that $M _{u} \in \Omega _{p} (E)\colon \ u = 1, 2$, and $M _{1}M
_{2} = M  ^{\prime }$, put $L ^{\prime } = M _{1} \cap M _{2}$.
We first show that $N(L ^{\prime }/E) = N(M _{1}/E)N(M _{2}/E)$
and $N(M ^{\prime }/E) = N(M _{1}/E) \cap N(M _{2}/E)$. In view of
(4.1) (ii), it suffices to consider the case where $L ^{\prime
} \neq E$. As $L ^{\prime }$ is $p$-quasilocal, (4.1) (ii)
yields $L ^{\prime \ast } = N(M _{1}/L ^{\prime })N(M
_{2}/L ^{\prime })$, so the equality $N(L ^{\prime }/E) =
N(M _{1}/E)N(M _{2}/E)$ is obtained from norm transitivity in the
towers $E \subset L ^{\prime } \subseteq M _{u}\colon \ u = 1,
2$. Suppose further that $[L ^{\prime }\colon E] = p ^{m}$ and fix
an extension $L$ of $E$ in $L ^{\prime }$ of degree $p$. As
in the proof of (4.1) (i), it is seen that an element $\l \in L
^{\ast }$ lies in $N(M _{i}/L)$, for some $i \in \{1, 2\}$, if and only if
$N _{E} ^{L} (\l) \in N(M _{i}/E)$. Since $[L ^{\prime
}\colon L] = p ^{m-1}$ and $L$ is $p$-quasilocal, this
makes it easy to prove inductively that $N(M ^{\prime }/E) = N(M _{1}/E)
\cap N(M _{2}/E)$. It follows from this result and (4.1) (i) that the natural
mapping $\Omega _{p} (E) \to {\rm Nr}(E)$ is injective. Thus the
statement that $E$ admits local $p$-CFT is proved, which allows us to
deduce Theorem 1.1 from Lemma 2.1 and Proposition 3.3.
\par
To finish the proof of Theorem 3.1 (and Theorem 1.2) we show that $E
^{\ast }/N(M/E) \cong \mathcal{G}(M/E) ^{d(p)}$, provided that $M \in
\Omega _{p} (E)$ and $\mathcal{G}(M/E)$ is noncyclic. Then ${\mathcal
G}(M/E)$ is an abelian $p$-group of rank $r(M) \geq 2$, so it follows
from Galois theory that $M/E$ has intermediate fields $\Phi _{1}$
and $\Phi _{2}$, such that $\Phi _{1}\Phi _{2} = M$, $\Phi _{1} \cap
\Phi _{2} = E$ and $\Phi _{2}/E$ is cyclic. This means that
$\mathcal{G}(M/E)$ is isomorphic to the direct product $\mathcal{G}(\Phi
_{1}/E) \times \mathcal{G}(\Phi _{2}/E)$, and $\mathcal{G}(\Phi _{1}/E)$
is of rank $r(M) - 1$ as a $p$-group. Since $N(\Phi _{1}/E)N(\Phi
_{2}/E) = E ^{\ast }$ and $N(\Phi _{1}/E) \cap N(\Phi _{2}/E) =
N(M/E)$, the natural diagonal embedding of $E ^{\ast }$ into $E
^{\ast } \times E ^{\ast }$ induces a group isomorphism
$E ^{\ast }/N(M/E) \cong E ^{\ast }/N(\Phi _{1}/E) \times E ^{\ast
}/N(\Phi _{2}/E)$. Now our proof is easily completed proceeding
by induction on $r(M)$.

\smallskip
\begin{rema}
For a PQL-field $E$, it follows from Theorem 3.1, Lemma 2.1 and
\cite{Ch6}, I, Lemma~4.2 (ii), that the natural map $\nu $ of
$\Omega (E)$ on the set $N _{\rm ab} (E) = \{N(M/E)\colon \ M \in
\Omega (E)\}$ satisfies (1.1). Therefore, for each $M \in \Omega
(E)$, the intersection ${\rm c}(M)$ of the fields $N \in \Omega
(E)\colon \ N(N/E) = N(M/E)$, lies in $\Omega (E)$, and we have
Br$(E) _{p} \neq \{0\}$ and $p \dagger [M\colon c(M)]$, for each
prime $p$ dividing $[{\rm c}(M)\colon E]$. Note also that $N({\rm
c}(M)/E) = N(M/E)$ and $N(M _{0}/E) \neq N(M/E)$ when $M _{0} \in
\Omega (E)$, $M _{0} \subseteq {\rm c}(M)$ and $M _{0} \neq {\rm
c}(M)$. Since the set Cl$(E) = \{{\rm c}(\Lambda )\colon \ \Lambda
\in \Omega (E)\}$ is closed under the formation of subextensions of
$E$, field intersections and finite compositums, whence $\nu $
induces a bijection of Cl$(E)$ on $N _{\rm ab} (E)$, these
observations allow us to consider ${\rm c}(M)$ as a class field of
$N(M/E)$.
\end{rema}

\section{Galois cohomological interpretation of Theorem 3.1}

\medskip
In this Section we consider some Galois cohomological aspects of the
problem of characterizing fields with LCFT. Let $p$ be a prime
number, $P$ an infinite pro-$p$-group, and $\FF _{p}$ a field with
$p$ elements. We say that $P$ is a $p$-group of Demushkin type, if
the (continuous cohomology) group homomorphism $\varphi _{h}\colon \
H ^{1} (P, \FF _{p}) \to H ^{2}(P,  \FF _{p})$ mapping each $g \in H
^{1} (P, \FF _{p})$ into the cup-product $h \cup g$ is surjective,
for every $h \in H ^{1} (P, \FF _{p}) \setminus \{0\}$. We call a
degree of $P$ the dimension of $H ^{2} (P, \FF _{p})$ as an $\FF
_{p}$-vector space. Local $p$-CFT and $p$-groups of Demushkin type
are related as follows:

\medskip
\begin{prop}
Let $E$ be a nonreal field containing a primitive $p$-th root of
unity, for some $p \in P(E)$. Then the following conditions are
equivalent:
\par
{\rm (i)} $E$ admits local $p$-{\rm CFT};
\par
{\rm (ii)} $\mathcal{G}(E (p)/E)$ is a $p$-group of Demushkin type
of degree $d \geq 1$;
\par
{\rm (iii)} $E$ is $p$-quasilocal with {\rm Br}$(E) _{p} \neq
\{0\}$;
\par
{\rm (iv)} $\mathcal{G}(E (p)/E)$ is of cohomological dimension two
and $_{p} {\rm Br}(E ^{\prime })$ is a trivial module over the
integral group ring $\ZZ [\mathcal{G}(E ^{\prime }/E)]$, for every
extension $E ^{\prime }$ of $E$ in $E (p)$ of degree $p$.
\end{prop}

\begin{proof}
The equivalence (i)$\leftrightarrow $(iii) is implied by Proposition
3.3, Remark 3.4 (ii) and Theorem 3.1. Note also that Br$(E) _{p} =
\{0\}$ if and only if $\mathcal{G}(E (p)/E)$ is a free
pro-$p$-group, i.e. a $p$-group of Demushkin type of degree zero
(cf. \cite{W1}, Theorem~3.1, \cite{W2}, page 725, and \cite{S1}, Ch.
I, 4.1 and 4.2). In particular, this holds when $\mathcal{G}(E
(p)/E)$ is of rank $1$ as a pro-$p$-group, since then $\mathcal{G}(E
(p)/E) \cong \ZZ _{p}$ (see, e.g., \cite{Ch6}, I, Remark~3.4 (ii)).
These observations, \cite{Ch6}, I, Lemma~3.8, and the end of
Proposition 2.2 (ii) prove that (ii)$\leftrightarrow $(iii). It
remains to be seen that (iii)$\leftrightarrow $(iv). As $E$ is
nonreal and $p \in P(E)$, one obtains from Galois theory and
\cite{Wh2}, Theorem~2, that $\mathcal{G}(E (p)/E)$ possesses a
closed normal subgroup $H$, such that $\mathcal{G}(E (p)/E)/H \cong
\ZZ _{p}$. Since Br$(E) _{p} \neq \{0\}$, this means that $H \neq
\{1\}$, so it follows from \cite{Ch6}, I, Proposition~4.6 (ii), and
Galois cohomology (see \cite{S1}, Ch. I, 4.2 and Proposition 15)
that $H$ and $\mathcal{G}(E (p)/E)$ have cohomological dimensions
cd$(H) = 1$ and cd$(\mathcal{G}(E (p)/E)) = 2$. Hence, by
Teichm\"uller's theorem (cf. \cite{Dr}, Ch. 9, Theorem 4) and
Proposition 2.2 (iii), (iii)$\to $(iv). To prove the implication
(iv)$\to $(iii), we need the following results of Galois cohomology
(see \cite{MS}, (11.5), and \cite{Ko1}, Proposition~3.26):

\medskip
(5.1) If $F$ is a field containing a primitive $p$-th root of unity,
and $M \subseteq F (p)$ is a finite Galois extension of $F$, then
there exists an isomorphism $\kappa _{M}\colon H ^{2} (\mathcal{G}(F
(p)/M), \FF _{p}) \cong $ $_{p} {\rm Br}(M)$ as $\ZZ
[\mathcal{G}(M/F)]$-modules. Also, the compositions Cor$_{M/F} \circ
\kappa _{M}$ and $\kappa _{F} \circ $ ${\rm cor}_{M/F}$ coincide,
where cor$_{M/F}$ is the corestriction map $H ^{2}(\mathcal{G}(F
(p)/M), \FF _{p}) \to H ^{2} (\mathcal{G}(F (p)/F), \FF _{p})$.
Hence, Cor$_{M/F}$ is surjective if and only if so is cor$_{M/F}$.

\medskip\noindent
It is sufficient to establish that Br$(E ^{\prime }/E) =$ $_{p} {\rm
Br}(E)$, for an arbitrary extension $E ^{\prime }$ of $E$ in $E
(p)$ of degree $p$. The equality cd$(\mathcal{G}(E (p)/E)) = 2$
ensures that Br$(E) _{p} \neq \{0\}$ and it follows from (5.1) and
\cite{NSW}, Proposition~3.3.8, that Cor$_{E'/E}$ maps $_{p} {\rm
Br}(E ^{\prime })$ surjectively on $_{p} {\rm Br}(E)$. We show that
$_{p} {\rm Br}(E ^{\prime })$ includes the preimage $\Pi _{p}$ of
$_{p} {\rm Br}(E)$ in Br$(E ^{\prime }) _{p}$ under Cor$_{E'/E}$.
Our assumptions guarantee (in conjunction with Galois theory and the
normality of maximal subgroups of finite $p$-groups, see \cite{L},
Ch. VIII and Ch. I, Sect. 6) that $E ^{\prime }/E$ is cyclic.
Observe also that Br$(E ^{\prime }) _{p}$ is a trivial
$\ZZ [\mathcal{G}(E ^{\prime }/E)]$-module. Indeed, let $\sigma $ be
a generator of $\mathcal{G}(E ^{\prime }/E)$, and let $b \in {\rm
Br}(E ^{\prime })$ be an element of order $p ^{k}$, for some $k \in
\NN $. Proceeding by induction on $k$, one sees that it
suffices to prove that $\sigma (b) = b$, provided that $k \ge 2$ and
$\sigma (b) = b + b _{0}$, for some $b _{0} \in _{p} {\rm Br}(E
^{\prime })$. The assumption on $b _{0}$ indicates that Cor$_{E'/E}
(b _{0}) = 0$, so it follows from (5.1) and \cite{LLMS}, Corollary,
that $b _{0} = a - \sigma (a)$, for some $a \in $ $_{p} {\rm Br}(E
^{\prime })$. This yields $\sigma (b + a) = b + a$, which implies
$\sigma (b) = b$ and our assertion about Br$(E ^{\prime }) _{p}$
(because $_{p} {\rm Br}(E ^{\prime })$ is a trivial $\ZZ
[\mathcal{G}(E ^{\prime }/E)]$-module). By Teichm\"uller's theorem,
the established property of Br$(E ^{\prime }) _{p}$ is equivalent to
the inclusion of Br$(E ^{\prime }) _{p}$ in the image of $\rho
_{E/E'}$. Since Cor$_{E'/E}$ maps $_{p} {\rm Br}(E ^{\prime })$
surjectively upon $_{p} {\rm Br}(E)$, it is now easy to obtain from
the RC-formula that $\Pi _{p} \subseteq $ $_{p} {\rm Br}(E ^{\prime
})$. Moreover, it becomes clear that the implication (iv)$\to $(iii)
will follow, if we show that $_{p} {\rm Br}(E)$ is included in the
subgroup $p{\rm Br}(E) _{p} = \{p\beta \colon \ \beta \in {\rm
Br}(E) _{p}\}$ of Br$(E) _{p}$. Denote by $B$ the extension of $E$
in $E (p)$ obtained by adjunction of a primitive $p ^{2}$-th root of
unity. It follows from \cite{MS}, (16.5), \cite{P}, Sect. 15.1,
Corollary b, and Kummer theory that $_{p} {\rm Br}(B) \subseteq
p{\rm Br}(B) _{p}$. Hence, the surjectivity of the homomorphism
$_{p} {\rm Br}(B) \to $ $_{p} {\rm Br}(E)$ induced by Cor$_{B/E}$
implies $_{p} {\rm Br}(E) \subseteq p{\rm Br}(E) _{p}$. Therefore,
Br$(E ^{\prime }/E) =$ $_{p} {\rm Br}(E)$ and (iv)$\to $(iii), so
Proposition 5.1 is proved.
\end{proof}

\begin{coro}
Let $E$ be an {\rm SQL}-field, such that ${\rm cd}_{p} (\mathcal{G}
_{E}) \neq 0$, for a given prime $p$. Then the Sylow
pro-$p$-subgroups of $\mathcal{G} _{E}$ are $p$-groups of Demushkin
type of degree $d _{p} \geq 1$, unless $p = {\rm char}(E)$ or $E$ is
formally real and $p = 2$.
\end{coro}

\begin{proof}
One may consider only the special case of $E  _{\rm sep} = E
(p)$ (see the proof of Proposition 3.6). Then Br$(E) _{p} \neq
\{0\}$ and our conclusion follows from Propositions 5.1, 2.2 (ii)
and \cite{S1}, Ch. II, Proposition 3.
\end{proof}

\medskip
It is easily seen that the degrees of $p$-groups of Demushkin type
are bounded by their ranks. The following conversion of this fact
can be deduced from \cite{MS}, (11.5), Proposition 5.1, \cite{Ch6},
I, Theorem 8.1, and the sufficiency part of (2.2) (by specifying
the cardinalities of the fields $E _{0}$ and $E$ in (2.2) (ii), see
\cite{Ch9}, Remark~5.4, for more details):

\medskip
(5.2) For any system $d \geq \aleph _{0}$ and $d _{p} \leq d\colon \
p \in \overline P$, of cardinal numbers, where $\overline P$ is the
set of prime numbers, there is a field $E$ containing a primitive
$p$-th root of unity, for every $p \in \overline P$, and such that
$\mathcal{G}(E (p)/E)$ and the Sylow pro-$p$-subgroups of
$\mathcal{G} _{E}$ are of rank $d$, Demushkin type and degree $d
_{p}$.

\medskip
By a Demushkin pro-$p$-group, we mean a $p$-group of Demushkin type
of degree $1$. Pro-$p$-groups $P$ of this kind and rank $r(P) \leq
\aleph _{0}$ have been classified by Demushkin, Labute and Serre
(cf. \cite{Lab1, Lab2} and further references there). When $r(P) =
\aleph _{0}$, by the main results of \cite{MW1} and \cite{MW2}, $P$
has $s$-invariant equal to zero if and only if $P \cong {\mathcal
G}(F (p)/F)$, for some field $F$. Hence, by applying Lemma 3.5 of
\cite{Ch4} (in a modified form adjusted to the case singled out by
\cite{MW2}, Proposition~3.1 (iii)), and arguing as in the proof of
(5.2), one supplements these results as follows:

\medskip
(5.3) For each sequence $G _{p}\colon \ p \in \overline P$, of
Demushkin pro-$p$-groups of rank $\aleph _{0}$ and $s$-invariant
zero, there exists a field $E$ such that $\mathcal{G}(E (p)/E)$ and
the Sylow pro-$p$-subgroups of $\mathcal{G} _{E}$ are isomorphic to
$G _{p}$ when $p$ ranges over $\overline P$.

\medskip
At the end of this Section, we show that the inequality $d \geq
\aleph _{0}$ in (5.2) is essential. This result is a special case of
\cite{Ku}, Corollary 2 (see also \cite{LLMS}, Sects. 1 and 2). For
convenience of the reader, we present it here with a short proof
based on Theorem 3.1 and Proposition 5.1.

\medskip
\begin{coro}
Let $E$ be a field containing a primitive $p$-th root of unity, for
some $p \in P(E)$, and with $\mathcal{G}(E (p)/E)$ a $p$-group of
Demushkin type of finite rank $r(p)$. Then $\mathcal{G}(E (p)/E)$ is
a Demushkin group or a free pro-$p$-group.
\end{coro}

\begin{proof}
In view of \cite{W2}, Lemma~7, and Proposition 5.1 (with its proof),
one may consider only the case where $r(p) \geq 3$ and Br$(E) _{p}
\neq \{0\}$. Take a field $E ^{\prime } \in \Omega (E)$ so that
$\mathcal{G}(E ^{\prime }/E)$ has exponent $p$ and order $p
^{r(p)-1}$. By Theorem 3.1 and \cite{MS}, (11.5), then $E ^{\ast p}
\subseteq N(E ^{\prime }/E)$ and $N(E ^{\prime }/E)$ is a subgroup
of $E ^{\ast }$ of index $p ^{(r(p)-1).d _{p}}$, so we have $(r(p) -
1).d _{p}$ $\leq r(p)$, where $d _{p}$ is the degree of
$\mathcal{G}(E (p)/E)$. In our case, this implies $d _{p} = 1$,
which proves Corollary 5.3.
\end{proof}

\medskip
Statement (5.2), Corollary 5.3 and \cite{Ch6}, II, Lemma 2.3, raise
interest in the latter part of (2.4) (ii), and in finding a
description of groups of Demushkin type realizable as Galois groups
of maximal $p$-extensions. Note also that (5.1) and Corollary 5.3
show that the equivalence (ii)$\leftrightarrow $(iv) in Proposition
5.1 generalizes the concluding assertion of \cite{LLMS}, Theorem~1
(independently of the elementary type conjecture formulated in
\cite{LLMS}). At the same time, (5.1), (5.2) and Proposition 5.1
indicate that this assertion is not necessarily true when
$\mathcal{G}(E (p)/E)$ has infinite rank as a pro-$p$-group.


\section{Preparation for the proof of Theorems 3.2 and 1.3}

Let $E$ be a field, $M \in \Omega (E)$, $\Pi $ the set of prime
divisors of $[M\colon E]$, and $M _{p} = M \cap E (p)$, for each $p
\in \Pi $. When $M \neq E$, the homomorphism group ${\rm Hom}
(E ^{\ast }, \mathcal{G}(M/E))$ is isomorphic to the direct group
product $\prod _{p \in \Pi } {\rm Hom} (E ^{\ast },$ $\mathcal{G}(M
_{p}/E))$, so Theorem 1.3 can be deduced from Theorem 3.2,
Lemma 2.1 and the primary tensor decomposition of cyclic
$E$-algebras (cf. \cite{P}, Sect. 15.3). The results of this Section
serve as a basis for the proof of Theorem 3.2, presented in Section
7. Our starting point are the following two lemmas.

\medskip
\begin{lemm} Let $E$ and $M$ be fields, such that $M \in \Omega
_{p} (E)$, for some $p \in P(E)$. Suppose that  $\mathcal{G}(M/E)$
has rank $t \geq 1$ as a $p$-group, and $F$ is an intermediate
field of $M/E$ of degree $[F\colon E] = p$. Then there exist cyclic
extensions $E _{1}, \dots , E _{t}$ of $E$ in $M$ with $E _{1} \dots
E _{t} = M$, $\prod _{i=1} ^{t} [E _{i}\colon E] = [M\colon E]$ and
$\prod _{i=1} ^{t} [(FE _{i})\colon F] = [M\colon F]$.
\end{lemm}

\begin{proof} In view of Galois theory, this is equivalent to the
following statement:

\medskip
(6.1) Let $G$ be a finite abelian $p$-group of rank $t \geq 1$ and
$H$ a maximal subgroup of $G$. Then these exist cyclic subgroups $G
_{1}, \dots , G _{t}$ of $G$, such that the (inner) products of
$G _{i}\colon \ i = 1,\dots , t$, and $H \cap G _{i}\colon \ i = 1,
\dots , t$, are direct and equal to $G$ and $H$, respectively.

\medskip
To prove (6.1) take a cyclic subgroup $G _{1}$ of $G$ of maximal
order so that the order of the group $H _{1} = H \cap G _{1}$
equals the exponent of $H$. The choice of $G _{1}$ ensures that $G
_{1}G _{0} = G$ whenever $G _{0}$ is maximal among the
subgroups of $G$, which trivially intersect $G _{1}$ (cf. \cite{F},
Sects. 15 and 27). This implies the existence of subgroups $H _{1}
^{\prime } \subseteq H$ and $G _{1} ^{\prime } \subseteq G$, such
that $H _{1} \cap H _{1} ^{\prime } = G _{1} \cap G _{1} ^{\prime }
= \{1\}$, $H _{1} ^{\prime } \subseteq G _{1} ^{\prime }$ and the
products $H _{1}H _{1} ^{\prime }$ and $G _{1}G _{1} ^{\prime }$
are equal to $H$ and $G$, respectively. Therefore, (6.1) can be
proved by induction on $t$.
\end{proof}

\smallskip
\begin{lemm}
Assume that $E$, $M$, $p$ and $t$ satisfy the conditions of Lemma
6.1, and let $F$ be a maximal subfield of $M$ including $E$. Then
there exist cyclic extensions $E _{1}, \dots , E _{t}$ of $E$ in
$M$, such that $\prod _{i=1} ^{t} [E _{i}\colon E] = [M\colon E]$,
$\prod _{i=1} ^{t} [(E _{i} \cap F)\colon E] = [F\colon E]$, and
the compositums $E _{1} \dots E _{t}$ and $(E _{i} \cap F) \dots (E
_{t} \cap F)$ are equal to $M$ and $F$, respectively.
\end{lemm}

\begin{proof}
This is equivalent to the following statement:

\medskip
(6.2) Let $G$ be a finite abelian $p$-group of rank $t \geq 1$, $H$ a
subgroup of $G$ of order $p$, and $\pi $ the natural homomorphism of
$G$ on $G/H$. Then $G$ contains elements $g _{1}, \dots , g _{t}$,
for which the products of the cyclic groups $\langle g _{1}\rangle
, \dots , \langle g _{t}\rangle $ and $\langle \pi (g _{1})\rangle
, \dots , \langle \pi (g _{t})\rangle $ are direct and equal to $G$
and $G/H$, respectively.

\medskip
For the proof of (6.2), consider a cyclic subgroup $C _{1}$ of $G$
of maximal order, and such that $C _{1} \cap H$ is of minimal
possible order. Then one can find a subgroup $C _{1} ^{\prime }$ of
$G$ so that $C _{1} \cap C _{1} ^{\prime } = \{1\}$, $C _{1}C _{1}
^{\prime } = G$ and $H \subset C _{1} \cup C _{1} ^{\prime }$. This
implies $G/H$ is isomorphic to the direct product $C _{1}H/H \times
C _{1} ^{\prime }H/H$, which allows one to prove (6.2) by induction
on $t$.
\end{proof}

\medskip
Let now $G$ be a finite abelian $p$-group of rank $t \geq 2$. A
subset $g = \{g _{1}, \dots , g _{t}\}$ of $G$ is called an ordered
basis of $G$, if $g$ is a basis of $G$ (i.e. the group product
$\langle g _{1}\rangle \dots \langle g _{t}\rangle $ is direct and
equal to $G$) and the orders of the elements of $g$ satisfy the
inequalities ${\rm o}(g _{1}) \geq \dots \geq {\rm o}(g _{t})$. It is known that
the automorphism group Aut$(G)$ acts transitively on the set
Ob$(G)$ of ordered bases of $G$. Arguing by induction on $t$, and
using the fact that cyclic subgroups of $G$ of exponent ${\rm o}(g _{1})$
are direct summands in $G$ (see \cite{F},  Sect. 27), one obtains
that Aut$(G)$ has the following system of generators:

\medskip
(6.3) Aut$(G) = \langle d(k, m; h), t(i, j, s; h)\colon \ h \in {\rm
Ob}(G)\rangle $, where $i, j, k, s$ and $m$ are integers with $1 \leq
k, i, j \leq t$, $i \neq j$, $1 < m < {\rm o}(h _{k})$, $1 \le s < {\rm o}(h
_{i})$, $\gcd {(m, {\rm o}(h _{k}))} = 1$ and max$\{1, {\rm o}(h
_{i})/{\rm o}(h _{j})\} \mid s$, and $d(k, m; h)$, $t(i, j, s; h)$ are
defined by the data $d(k, m; h) (h _{k}) = h _{k} ^{m}$, $d(k,m; h) (h
_{k'}) = h _{k'}\colon \ k ^{\prime } \neq k$, and $t(i, j, s; h) (h
_{j}) = h _{j}h _{i} ^{s}$, $t(i, j, s; h)$ $(h _{j'}) = h
_{j'}\colon \ j ^{\prime } \neq j$.

\medskip \noindent
This allows us to prove the following two lemmas without
encountering serious technical difficulties.


\begin{lemm}
Let $E$, $M$, $p$ and $t$ satisfy the conditions of Lemma 6.1, and
let $E$ be a $p$-quasilocal field. Suppose that $B$ a cyclic
subgroup of $ \ {\rm Br}(E) _{p}$ of order ${\rm o}(B)$ divisible by
$[M\colon E]$, $b$ is a generator of $B$, and $B(M/E)$ is the group
of those $\b \in E ^{\ast }$, for which $[(L/E, \tau , \b )] \in B$ whenever
$L/E$ is a cyclic extension, $L \subseteq M$ and $\langle \tau \rangle
=  \mathcal{G}(L/E)$. Assume also that $E _{1}, \dots , E _{t}$ are
cyclic extensions of $E$ in $M$, such that $E _{1} \dots E _{t} = M$
and $[M\colon E] = \prod _{j=1} ^{t} [E _{j}\colon E]$, and for each
index $j \leq t$, let $E _{j} ^{\prime }$ be the compositum of
the fields $E _{i}\colon \ i \neq j$, $\tau _{j}$ a generator of
$\mathcal{G}(E _{j}/E)$, and $\s _{j}$ the unique $E _{j} ^{\prime
}$-automorphism of $M$ extending $\tau _{j}$. Then there exist a
group homomorphism $\omega _{M/E,b}\colon \ B(M/E) \to {\mathcal
G}(M/E)$, and elements $c _{1}, \dots , c _{t}$ of $E ^{\ast }$ with
the following properties:

{\rm (i)} $c _{j} \in N(E _{j} ^{\prime }/E)$ and $[(E _{j}/E, \tau _{j}, c
_{j})] = [{\rm o}(B)/[E _{j}\colon
E]].b$, for each index $j$; the co-set $c _{j}N(M/E)$ is uniquely
determined by $b$ and $\tau _{j}$;

{\rm (ii)} $\omega _{M/E,b}$ is the unique homomorphism of
$B(M/E)$ on $\mathcal{G}(M/E)$ mapping $c _{j}$ into $\s _{j}\colon
\ j = 1, \dots , t$, and with a kernel equal to $N(M/E)$;

{\rm (iii)} $\omega _{M/E, b}$ does not depend on the choice of
the $t$-tuples $(E _{1}, \dots , E _{t})$ and $(\tau _{1}, \dots ,
\tau _{t})$; it induces a group isomorphism
$B(M/E)/N(M/E) \cong \mathcal{G}(M/E)$.
\end{lemm}


\begin{proof}
If $t = 1$, our assertions can be deduced from Theorem
3.1 and the general theory of cyclic algebras (see \cite{P}, Sect.
15.1). Henceforth, we assume that $t \geq 2$. Using consecutively
Galois theory and Theorem 3.1, one obtains that $N(E _{j}/E)N(E _{j}
^{\prime }/E) = E ^{\ast }\colon \ j = 1, \dots , t$, which implies
the existence of elements $c _{1}, \dots , c _{t}$ of $E ^{\ast }$
with the properties required by Lemma 6.3 (i). Denote by $T(M/E)$
the subgroup of $E ^{\ast }$ generated by the set $N(M/E) \cup \{c
_{1}, \dots , c _{t}\}$. It is easily verified that $\s _{1},
\dots , \s _{t}$ and $c _{1}N(M/E), \dots , c _{t}N(M/E)$ form
bases of the groups $\mathcal{G}(M/E)$ and $T(M/E)/$ $N(M/E)$,
respectively. Therefore, there exists a unique homomorphism $\omega
_{M/E, b}$ of $T(M/E)$ on $\mathcal{G}(M/E)$, such that $\omega _{M/E,
b} (c _{j}) = \s _{j}\colon \ j = 1, \dots , t$, and ${\rm
Ker}(\omega _{M/E, b}) = N(M/E)$. The mapping $\omega _{M/E,
b}$ is surjective, so it induces canonically an isomorphism of
$T(M/E)/N(M/E)$ on $\mathcal{G}(M/E)$. We aim at proving that $T(M/E) =
B(M/E)$. Assuming that $[M\colon E] = p ^{m}$, and proceeding by
induction on $m$, one obtains that this can be deduced from Lemma
6.2 and \cite{P}, Sect. 15.1, Corollary~b, if $\omega _{M/E, b}$ has
the property claimed by the former part of Lemma 6.3 (iii). In
order to establish this property (the crucial point in our proof),
consider another basis $\s _{1} ^{\prime }, \dots , \s _{t}
^{\prime }$ of $\mathcal{G}(M/E)$, and for each $j \in \{1, \dots ,
t\}$, let $H _{j}$ be the subgroup of $\mathcal{G}(M/E)$ generated by
the elements $\s _{i} ^{\prime }\colon \ i \neq j$, $F _{j}$ the
fixed field of $H _{j}$, $\tau _{j} ^{\prime }$ the $E$-automorphism of
$F _{j}$ induced by $\s _{j} ^{\prime }$, and $F _{j} ^{\prime }$ the
compositum of all $F _{i}$ with $i \neq j$. It follows from Galois theory
that $F _{1} \dots F _{t} = M$, $\prod _{i=1} ^{t} [F _{i}\colon E] = [M\colon
E]$, $F _{j}/E$ is a cyclic extension, $\mathcal{G}(F _{j}/E) = \langle \tau
_{j} ^{\prime }\rangle $, and $\s _{j} ^{\prime }$ is the unique
$F _{j} ^{\prime }$-automorphism of $M$ extending $\tau _{j}
^{\prime }$ ($j = 1, \dots , t$). Since $\omega _{M/E, b}$ is
surjective, it maps some elements $c _{1} ^{\prime }, \dots , c _{t}
^{\prime }$ of $T(M/E)$ into $\s _{1} ^{\prime }, \dots , \s
_{t} ^{\prime }$, respectively. Applying Lemma 6.2 and \cite{P},
Sect. 15.1, Corollary~b, one concludes that the proof of Lemma 6.3
will be complete, if we show that $c _{j} ^{\prime }\in N(F _{j}
^{\prime }/E)$ and $[(F _{j}/E, \tau _{j} ^{\prime }, c _{j}
^{\prime })] = [{\rm o}(B)/[F _{j}\colon E]].b$, for every index $j$.
\par
It is clearly sufficient to consider the special case in which
$\{\s _{1}, \dots , \s _{t}\}$ and $\{\s _{1} ^{\prime
}, \dots , \s _{t} ^{\prime }\}$ are ordered bases of ${\mathcal
G}(M/E)$ (which implies $[E _{\rho }\colon E] = [F _{\rho }\colon
E]$, $\rho = 1, \dots , t$). In view of \cite{P}, Sect. 15.1,
Corollary~a, and (6.3), one may assume in addition that $\s _{j}
^{\prime } = \s _{j}\s _{i} ^{s}$, $c _{j} ^{\prime } = c
_{j}c _{i} ^{s}$, $\s _{u} ^{\prime } = \s _{u}$ and $c _{u}
^{\prime } = c _{u}\colon \ u \neq j$, for some pair $(i, j)$ of
different indices, and some integer $s$ satisfying the inequalities
$1 \le s < {\rm o}(\s _{i})$ and divisible by max$ \{1, {\rm o}(\s _{i})/{\rm o}(\s
_{j})\}$.
Then it follows from Galois theory that $F _{i}F _{j} = E _{i}E
_{j}$, $F _{w} = E _{w}, \tau _{w} ^{\prime } = \tau _{w}\colon \ w
\neq i$, and $F _{u} ^{\prime } = E _{u} ^{\prime }\colon \ u \neq
j$. One also sees that if $t \geq 3$ and $y \in (\{1, \dots ,t\}
\setminus \{i, j\})$, then $c _{j} ^{\prime } \in N(E _{y}/E)$.
Since the cyclic $E$-algebras $(E _{j}/E, \tau _{j}, c _{j})$ and $(E
_{j}/E, \tau _{j}, c _{j} ^{\prime })$ are isomorphic (see \cite{P},
Sect. 15.1, Proposition~b), and by Lemma 4.1, $N(F _{j} ^{\prime
}/E) = \cap _{u \neq j} N(F _{u}/E)$, this reduces the proof of
Lemma 6.3 (ii) to the one of the assertions that $c _{j} ^{\prime }
\in N(F _{i}/E)$ and there is an $E$-isomorphism $(E _{i}/E, \tau
_{i}, c _{i}) \cong (F _{i}/E, \tau _{i} ^{\prime }, c _{i})$. We
first show that $(E _{i}/E, \tau _{i}, c _{i}) \cong (F _{i}/E, \tau
_{i} ^{\prime }, c _{i})$. The assumptions on $E _{i}, F _{i}$ and
$E _{j} = F _{j}$ guarantee that the field $E _{i}E _{j} = F _{i}E
_{j}$ is isomorphic as an $E$-algebra to $E _{i} \otimes _{E} E
_{j}$ and $F _{i} \otimes _{E} E _{j}$ (cf. \cite{P}, Sect. 14.7,
Lemma~b). It is therefore easy to see from the general properties of
tensor products (cf. \cite{P}, Sect. 9.2, Proposition~c) that $(E
_{i}/E, \tau _{i}, c _{i}) \otimes _{E} (E _{j}/E, \tau _{j}, c
_{j}) \cong (F _{i}/E, \tau _{i} ^{\prime }, c _{i}) \otimes _{E} (E
_{j}/E, \tau _{j}, c _{j}c _{i} ^{s'})$ as $E$-algebras, where $s
^{\prime } = s.[{\rm o}(\s _{j})/{\rm o}(\s _{i})]$. Since $s ^{\prime }
\in \ZZ $ and $c _{i} \in N(E _{j}/E)$, there exists an
$E$-isomorphism $(E _{j}/E, \tau _{j}, c _{j})$ $\cong (E _{j}/E, \tau
_{j}, c _{j}c _{i} ^{s'})$. The obtained results prove that $(E
_{i}/E, \tau _{i}, c _{i})$ and $(F _{i}/E,$ $\tau _{i} ^{\prime }, c
_{i})$ are similar over $E$. As $[E _{i}\colon E] = [F _{i}\colon
E]$, we also have $[(E _{i}/E, \tau _{i}, c _{i})\colon E]
= [(F
_{i}/E, \tau _{i} ^{\prime }, c _{i})\colon E]$ $= [E _{i}\colon E]
^{2}$, so it turns out that $(E _{i}/E, \tau _{i}, c _{i}) \cong (F
_{i}/E,$ $\tau _{i} ^{\prime }, c _{i})$, as claimed.

\par
It remains to be seen that $c _{j} ^{\prime } \in N(F _{i}/E)$.
Suppose first that $F _{i} \cap E _{i} = E$, take elements $\b
_{i} \in E _{j}$, $\b _{j} \in E _{i}$, $\d \in {\rm Br}(E)$
and an algebra $D \in d(E)$ so that $N _{E} ^{E _{j}} (\b _{i}) =
c _{i}$, $N _{E} ^{E _{i}} (\b _{j}) = c _{j}$, $([E _{i}\colon
E].[E _{j}\colon E])\d = b$, and ${\rm o}(B)\d = [D]$, and denote
by $\varphi _{i}$ and $\varphi _{j}$ the automorphisms of $E _{i}E
_{j}$ induced by $\s _{i}$ and $\s _{j}$, respectively. It
is easily verified that $[E _{j}\colon E].[D] = [(E _{i}/E, \tau
_{i}, c _{i})]$ and $\mathcal{G}(E _{i}E _{j}/E _{j}) = \langle \varphi
_{i}\rangle $. This, combined with the fact that $E _{i}E _{j} = F
_{i}E _{j}$, $E _{i} \cap E _{j} = E$ and $\varphi _{i}$ extends
$\tau _{i}$, enables one to deduce from (4.1) (iii), Proposition 2.2
and the RC-formula that $D \otimes _{E} E _{j}$ is similar to the
cyclic $E _{j}$-algebra $(E _{i}E _{j}/E _{j}, \varphi _{i}, \b
_{i})$. Since $[E _{i}\colon E].[D] = [(E _{j}/E, \tau _{j}, c
_{j})]$, $\mathcal{G}(E _{i}E _{j}/E _{i}) = \langle \varphi
_{j}\rangle $, and $\varphi _{j}$ is a prolongation of $\tau _{j}$,
it is analogously proved that $[D \otimes _{E} E _{i}] = [(E _{i}E
_{j}/E _{i}, \varphi _{j}, \b _{j})]$ in Br$(E _{i})$. At the
same time, it is clear from Galois theory and the condition $F _{i}
\cap E _{i} = E$ that $\mathcal{G}(E _{i}E _{j}/E) = \langle \varphi
_{j}, \varphi _{i} ^{s}\rangle $ (i.e. ${\rm g.c.d.}(s, p) = 1$),
$[E _{i}\colon E] \leq [E _{j}\colon E]$, and $E _{i}F _{i} = E
_{i}\Phi $, where $\Phi $ is the extension of $E$ in $E _{j}$ of
degree $[\Phi \colon E] = [E _{i}\colon E]$. Note also that
$\varphi _{j}$ extends $\tau _{i} ^{\prime -s}$, since $\s _{j}
^{\prime } (\l ) = \l $, for every $\l \in F _{i}$.
Observing now that $[(E _{i}/E, \tau _{i}, c _{i})] = \omega .[(E
_{j}/E, \tau _{j}, c _{j})]$ and $[(E _{i}F _{i}/E _{i}, \psi _{j},
\b _{j})] = \omega .[(E _{i}E _{j}/E _{i}, \varphi _{j}, \b
_{j})]$, where $\omega = [E _{j}\colon E]/[E _{i}\colon E] = [E
_{i}E _{j}\colon E _{i}F _{i}]$ and $\psi _{j}$ is the automorphism
of $E _{i}F _{i}$ induced by $\varphi _{j}$ (cf. \cite{P}, Sect.
15.1, Corollary~b), one obtains from (4.1) (iii) that $(E _{i}/E,
\tau _{i}, c _{i}) \cong (\Phi /E, \bar \psi _{j}, c _{j}) \cong (F
_{i}/E, \tau _{i} ^{\prime -s}, c _{j})$ over $E$ ($\bar \psi _{j}$
being the automorphism of $\Phi $ induced by $\varphi _{j}$). Since
$(E _{i}/E, \tau _{i}, c _{i}) \cong (F _{i}/E, \tau _{i} ^{\prime
}, c _{i})$, these results indicate that $(F _{i}/E, \tau _{i}
^{\prime -s}, c _{j}) \cong (F _{i}/E, \tau _{i} ^{\prime -s}, c
_{i} ^{-s})$, which means that $c _{j} ^{\prime } \in N(F _{i}/E)$.
\par
Let now $E _{i} \cap F _{i} = \widetilde E$, $[\widetilde E\colon E]
= \mu > 1$, $s/\mu = \tilde s$ and $E _{i}E _{j} =  E _{i,j}$. Then
$E _{i,j} \in \Omega (\widetilde E)$ and $\mathcal{G}(E
_{i,j}/\widetilde E) = \mathcal{G}(E _{i,j}/E _{i})\mathcal{G}(E _{i,j}/F
_{i})$, which implies $\tilde s \in \ZZ $ and ${\rm
g.c.d.}(\tilde s, p) = 1$. For each index $u \neq i$, take $\g
_{u} \in E _{u} ^{\prime }$ so that $N _{E} ^{E _{u}'} (\g _{u})
= c _{u}$, and put $\tilde c _{u} = N _{\widetilde E} ^{E _{u}'}
(\g _{u})$, $\widetilde E _{u} = E _{u}\widetilde E$ and
$\widetilde F _{u} = F _{u}\widetilde E$. Since $E _{i} \cap E _{u}
= E$, it follows from (4.1) (ii) that $E _{i} \cap \tilde E
_{u} = \widetilde E$, $[\widetilde E _{u}\colon \widetilde E] = [E
_{u}\colon E]$, $\tau _{u}$ uniquely extends to a $\widetilde
E$-automorphism $\tilde \tau _{u}$ of $\widetilde E _{u}$, and
$\mathcal{G}(\widetilde E _{u}/\widetilde E) = \langle \tilde \tau
_{u}\rangle $. Note also that $N _{E} ^{E _{u}} (\th _{u}) = N
_{\widetilde E} ^{\widetilde E _{u}} (\th _{u})$, $\th _{u}
\in E _{u}$, whence $c _{i} \in N(\widetilde E _{u}/\widetilde E)$.
Fix elements $b _{\mu } \in {\rm Br}(E)$ and $\Delta _{u} \in d(E)$
so that $\mu .b _{\mu } = b$ and $[\Delta _{u}] = [{\rm o}(B)/[E _{u}\colon
E]]b _{\mu }$, and put $b ^{\prime } =  \rho _{E/\widetilde E} (b
_{\mu })$. It is clear from the double centralizer theorem (cf.
\cite{P}, Sects. 12.7 and 13.3) that $[(E _{i}/E, \tau _{i}, c _{i})
\otimes _{E} \widetilde E] = [(E _{i}/\widetilde E, \tau _{i} ^{\mu },
c _{i})]$ in Br$(\widetilde E)$. Applying Proposition 2.2, (4.1) (iii)
and the RC-formula, one obtains consecutively that ${\rm o}(B)$
equals the order of $b ^{\prime }$ in Br$(\widetilde E)$, $[(E
_{i}/E, \tau _{i}, c _{i})] = [{\rm o}(B)/[E _{i}\colon \widetilde E]]b _{\mu
}$, $[(E _{i}/\widetilde E, \tau _{i} ^{\mu }, c _{i})] = [{\rm o}(B)/[E
_{i}\colon \widetilde E]].b ^{\prime }$, and for each $u \neq i$,
$[(E _{u}/E, \tau _{u}, c _{u})] = \mu [\Delta _{u}]$ and
$[(\widetilde E _{u}/\widetilde E, \tilde \tau _{u}, \tilde c _{u})] =
[\Delta _{u} \otimes _{E} \widetilde E] = [{\rm o}(B)/[\widetilde E
_{u}\colon \widetilde E]].b ^{\prime }$. Consider now
$M/\widetilde E$, $\tilde s$, $b ^{\prime }$, the fields $E _{i}, F
_{i}$, $\widetilde E _{u}, \widetilde F _{u}\colon \ u \neq i$, and
the algebras $(E _{i}/\widetilde E, \tau _{i} ^{\mu }, c _{i})$, $(F
_{i}/\widetilde E, \tau _{i} ^{\prime \mu }, c_{i})$, $(\widetilde E
_{j}/\widetilde E, \tilde \tau _{j}, \tilde c _{j})$, instead of $M/E$,
$s$, $b$, $E _{u'}, F _{u'}\colon \ u ^{\prime } = 1, \dots , t$, and
$(E _{i}/E, \tau _{i}, c _{i})$, $(F _{i}/E, \tau _{i} ^{\prime }, c
_{i})$, $(E _{j}/E, \tau _{j},$ $c _{j})$, respectively. As in the
proof of our assertion in case ${\rm g.c.d.}(s, p) = 1$, one
concludes that $\tilde c _{j}c _{i} ^{\tilde s} \in N(F _{i}/\widetilde
E)$. Since $N _{E} ^{\widetilde E} (\tilde c _{j}c _{i} ^{\tilde s}) = c
_{j} ^{\prime }$, this yields $c _{j} ^{\prime } \in N(F _{i}/E)$ (and
$T(M/E) = B(M/E)$), which completes the proof of Lemma 6.3.
\end{proof}

\begin{lemm}
In the setting of Lemma 6.3, let $F$ be an extension of $E$ in $M$,
and $\pi _{M/F}$ the natural projection of $\mathcal{G}(M/E)$ on
${\mathcal G}(F/E)$. Also, let $\rho _{E/F} (b) = b ^{\prime }$, $B
^{\prime } = \langle b ^{\prime }\rangle $, and $B _{\k } = \langle b
_{\k }\rangle $, where $b _{\k } = \k .b$, for some divisor $\k $ of
${\rm o}(B)/[M\colon E]$. Then $[M\colon E]$ and $[M\colon F]$
divide ${\rm o}(B _{\k })$ and ${\rm o}(B ^{\prime })$,
respectively, and the mappings $\omega _{M/E, b}$, $\omega _{M/E, b
_{\k }}$, $\omega _{F/E, b}$ and $\omega _{M/F, b'}$, determined
in accordance with Lemma 6.3, are related as follows:

{\rm (i)} $\omega _{M/E, b} = \omega _{M/E, b _{\k }}$ and
$\omega _{F/E, b} =  \pi _{M/F} \circ \omega _{M/E, b}$;

{\rm (ii)} $\omega _{M/F, b'} (\g ) = \omega _{M/E, b} (N _{E}
^{F} (\g ))$, for every $\g \in F ^{\ast }$.
\end{lemm}

\begin{proof}
Clearly, ${\rm o}(B _{\k }) = {\rm o}(B)/\k $, and
Proposition 2.2 implies that ${\rm o}(B ^{\prime }) = {\rm
o}(B)/[F\colon E]$, so the assertions that $[M\colon E] \mid {\rm
o}(B _{\k })$, $[M\colon F] \mid {\rm o}(B ^{\prime })$ and $\omega
_{M/E, b} = \omega _{M/E, b _{\k }}$ are obvious. Note also that if
$t = 1$, then the remaining statements of the lemma can be deduced
from the general properties of cyclic algebras. Suppose further that
$t \geq 2$. First we prove that $\omega _{F/E, b} = \pi _{M/F} \circ
\omega _{M/E, b}$ in case $F$ is a maximal subfield of $M$. Let $E
_{1}, \dots , E _{t}$ be extensions of $E$ in $M$ with the
properties described by Lemma 6.2. Then there exists an index $j$,
such that $E _{j} \cap F$ is a maximal subfield of $E _{j}$ and $E
_{i} \subseteq F$, for any other index $i$. This allows us to prove
our assertion by applying Lemma 6.3 and \cite{P}, Sect. 15.1,
Corollary~b.

\par
We turn to the proof of Lemma 6.4 (ii) in the special case of
$[F\colon E] = p$. Our argument goes along the same lines as the
concluding part of the proof of Lemma 6.3, so we present it briefly.
Take $E _{1}, E _{1} ^{\prime }, \dots , E _{t}, E _{t} ^{\prime }$,
$\tau _{1}, \s _{1}, \dots , \tau _{t}, \s _{t}$ and $c _{1}, \dots
, c _{t}$ in accordance with Lemmas 6.1 and 6.3. Then $F \subseteq E
_{j}$ and $E _{u} \cap F = E\colon \ u \neq j$, for some index $j$.
This implies that $F \subseteq E _{u} ^{\prime }$ and $\tau _{u}$
uniquely extends to an $F$-automorphism $\tilde \tau _{u}$ of the
field $E _{u}F := \tilde E _{u}$, when $u$ runs across the set $W
_{j} = \{1, \dots , t\} \setminus \{j\}$. Choose elements $\a _{1}
\in E _{1} ^{\prime }, \dots , \a _{t} \in E _{t} ^{\prime}$ so that
$N _{E} ^{E _{k}'} (\a _{k}) = c _{k}\colon \ k = 1, \dots ,t$, and
put $\tilde E _{j} = E _{j}$, $\tilde c _{j} = c _{j}$, $\tilde \tau
_{j} = \tau _{j} ^{p}$, $\tilde \s _{j} = \s _{j} ^{p}$, and $\tilde
c _{u} = N _{F} ^{E _{u}'} (\a _{u}), \tilde \s _{u} = \s _{u}\colon
\ u \in W _{j}$. It is easily verified that $N _{E} ^{F} (\tilde c
_{j}) = c _{j} ^{p}$ and $N _{E} ^{F} (\tilde c _{u}) = c _{u}$, $u
\in W _{j}$. Therefore, it follows from (4.1) (iii), \cite{P}, Sect.
14.7, Lemma~a, and the equality ${\rm o}(B ^{\prime }) = {\rm
o}(B)/p$ that $M/F$, $b ^{\prime }$, and $\tilde E _{\rho }$,
$\tilde \tau _{\rho }, \tilde \s _{\rho }, \tilde c _{\rho }$, where
$\rho $ runs through $W _{j}$ or $\{1, \dots , t\}$ depending on
whether or not $F = E _{j}$, are related as in Lemma 6.3. Since the
RC-formula and \cite{P}, Sect. 15.1, Proposition~b, yield Cor$_{F/E}
([(E _{j}/F, \tau _{j} ^{p}, c _{j})]) = [(E _{j}/E, \tau _{j}, c
_{j} ^{p})]$, this proves Lemma 6.4 (ii) in the case of $[F\colon E]
= p$.

Let finally $F$ be any proper extension of $E$ in $M$ different
from $M$. By Galois theory and the structure of finite abelian
groups, then $M$ possesses subfields $F _{0}$ and $M _{0}$, such
that $E \subset F _{0} \subseteq F \subseteq M _{0}$ and $[F
_{0}\colon E] = [M\colon M _{0}] = p$. In view of the transitivity
of norm mappings, canonical projections of Galois groups and
scalar extension maps of Brauer groups in towers of intermediate
fields of $M/E$ (cf. \cite{P}, Sect. 9.4, Corollary~a), the
considered special cases of Lemma 6.4 enable one to complete
inductively its proof.
\end{proof}



\section{Existence and form of Hasse symbols}


The main purpose of this Section is to prove Theorem 3.2. Fix a
generator $\varphi _{\infty }$ of $\mathcal{G}(E _{\infty }/E)$ as a
topological group, and take a subset $\Lambda _{p} (E) = \{b _{i,n}
(p)\colon \ i \in I _{p}, n \in \NN \}$ of Br$(E) _{p}$ so
that $\{b _{i,1} (p)\colon \ i \in I _{p}\}$ is a basis of $_{p}
{\rm Br}(E)$ as an $\FF _{p}$-vector space, and $pb _{i,n} (p) = b
_{i,n-1} (p)\colon \ i \in I _{p}, n \in \ZZ $ and $n \geq 2$ (the
existence of $\Lambda _{p} (E)$ is implied by Proposition 2.2 (ii)).
For each $E ^{\prime } \in \Omega _{p} (E)$, let $\rho _{E/E'} (b
_{i,n} (p)) = b _{i,n} (E ^{\prime })\colon \ (i, n) \in I _{p}
\times \NN $, $E ^{\prime } _{\infty } = E _{\infty }E
^{\prime }$, and $[(E ^{\prime } \cap E _{\infty })\colon E] = m(E
^{\prime })$. As $E _{\infty }/E$ is a $\ZZ _{p}$-extension and $\ZZ
_{p}$ is isomorphic to its open subgroups (see \cite{S1}, Ch. I,
1.5, 4.2), it follows from Galois theory that $E _{\infty } ^{\prime
}/E ^{\prime }$ is a $\ZZ _{p}$-extension and $\varphi _{\infty }
^{m(E')}$ uniquely extends to an $E ^{\prime }$-automorphism
$\varphi _{\infty } (E ^{\prime })$ of $E _{\infty} ^{\prime}$; one
also sees that $\varphi _{\infty } (E ^{\prime })$ topologically
generates $\mathcal{G}(E _{\infty } ^{\prime }/E ^{\prime })$. For
each $n \in \NN $ and $i \in I _{p}$, denote by $\varphi _{n} (E
^{\prime })$ the automorphism of $\Gamma _{n}E ^{\prime }$ induced
by $\varphi _{\infty } (E ^{\prime })$, and by $g _{i,n} (E ^{\prime
})$ the element of $\mathcal{G}(\Gamma _{n}E ^{\prime }/E ^{\prime
}) ^{d(p)}$ with components $g _{i,n} (E ^{\prime }) _{i} = \varphi
_{n} (E ^{\prime })$, and $g _{i,n} (E ^{\prime }) _{i'} = 1\colon \
i ^{\prime } \in I _{p} \setminus \{i\}$. By Proposition 2.2 and
\cite{P}, Sect. 15.1, Proposition~a, $E ^{\prime \ast }$ has a
subset $C _{p} (E ^{\prime }) = \{c _{i,n} (E ^{\prime })\colon \ i
\in I _{p}, n \in \NN \}$, such that $[((\Gamma _{n}E ^{\prime })/E
^{\prime }, \varphi _{n} (E ^{\prime }), c _{i,n} (E ^{\prime }))] =
b _{i,n} (E ^{\prime })$, for each $(i, n) \in I _{p} \times \NN $.
Observe that, for any $n \in \NN $, there is a unique surjective
homomorphism $( \ , (\Gamma _{n}E ^{\prime })/E ^{\prime })\colon \
E ^{\prime \ast } \to \mathcal{G}(\Gamma _{n}E ^{\prime }/E ^{\prime
}) ^{d(p)}$, whose kernel is $N(\Gamma _{n}E ^{\prime }/E ^{\prime
})$, and which maps $c _{i,n} (E ^{\prime })$ into $g _{i,n} (E
^{\prime })$, when $i \in I _{p}$. Fix some $M ^{\prime } \in \Omega
_{p} (E ^{\prime })$ and $\mu  \in \ZZ $ so that $p ^{\mu } \geq [M
^{\prime }\colon E]$, and for each $i \in I _{p}$, let $B(M ^{\prime
}/E ^{\prime }) _{i} = \{\b _{i} \in E ^{\prime \ast }\colon \ [(L
^{\prime }/E ^{\prime }, \s ^{\prime }, \b _{i})] \in \langle b
_{i, \mu } (E ^{\prime })\rangle \}$, for every cyclic extension $L
^{\prime }$ of $E ^{\prime }$ in $M ^{\prime }$, where $\s ^{\prime
}$ is a generator of $\mathcal{G}(L ^{\prime }/E ^{\prime })$. It
follows from Proposition 2.2 (ii)-(iii), the definition of $\Lambda
_{p} (E)$ and Lemma 6.3 that the groups $\overline B(M ^{\prime }/E
^{\prime }) _{i} := B(M ^{\prime }/E ^{\prime }) _{i}/N(M ^{\prime
}/E ^{\prime })$, $i \in I _{p}$, have the following property:

\medskip
(7.1) $\overline B(M ^{\prime }/E ^{\prime }) \cong \mathcal{G}(M
^{\prime }/E ^{\prime })$, for each index $i$, and the inner product
of $\overline B(M ^{\prime }/E ^{\prime }) _{i}\colon \ i \in I
_{p}$, is direct and coincides with $E ^{\prime \ast }/N(M ^{\prime
}/E ^{\prime })$.

\par
\medskip \noindent
Therefore, there exists a unique homomorphism $( \ , M ^{\prime }/E
^{\prime })$ of $E ^{\prime \ast }$ into

\noindent
$\mathcal{G}(M ^{\prime }/E ^{\prime
}) ^{d(p)}$, mapping $B(M ^{\prime }/E ^{\prime }) _{i}$ into the
$i$-th component of $\mathcal{G}(M ^{\prime }/E ^{\prime }) ^{d(p)}$
by the formula $\b _{i} \to  \omega _{M'/E',b _{i,\mu } (E')} (\b
_{i})$, for each $i \in I _{p}$, where $\omega _{M'/E',b _{i,\mu }
(E')}$ is defined as in Lemma 6.3. Hence, ${\rm Ker}( \ ,M ^{\prime
}/E ^{\prime }) = N(M ^{\prime }/E ^{\prime })$, and by Lemma 6.4,
the sets $H(E ^{\prime }) = \{( \ , M ^{\prime }/E ^{\prime })\colon
M ^{\prime } \in \Omega _{p} (E ^{\prime })\}$, $E ^{\prime } \in
\Omega _{p} (E)$, consist of surjections related as required by
Theorem 3.2 (ii)$\div $(iii).

\par
Suppose now that $\Theta _{p} (E ^{\prime }) = \{\th (M ^{\prime
}/E ^{\prime })\colon \ E ^{\prime \ast } \to \mathcal{G}(M ^{\prime
}/E ^{\prime }) ^{d(p)}, M ^{\prime } \in \Omega _{p} (E ^{\prime
})\}$, $E ^{\prime } \in \Omega _{p} (E)$, is a system of surjective
homomorphisms with the same kernels and relations, and such that
$\th (\Gamma _{n}/E) = ( \ , \Gamma _{n}/E)$, for every $n \in
\NN $. Then it follows from Proposition 2.2, the RC-formula
and (4.1) (iii) that $\th (\Gamma _{n}E ^{\prime }/E ^{\prime })
= ( \ , (\Gamma _{n}E ^{\prime })/E ^{\prime })$, for each pair $(E
^{\prime }, n) \in \Omega _{p} (E) \times \NN $. We show
that $\Theta _{p} (E ^{\prime }) = H _{p} (E ^{\prime })$, for any
$E ^{\prime } \in \Omega _{p} (E)$. This is obvious, if $E _{\infty
} = E (p)$, so we assume further that $E (p) \neq E _{\infty }$. In
view of Proposition 2.2 (i) and the already established part of
Theorem 3.2, it is sufficient to prove that $\th (M/E) =
( \ , M/E)$, for an arbitrary fixed field $M \in \Omega _{p} (E)$.
Note first that the compositum $ME _{\infty }$ possesses a subfield
$M _{0} \in \Omega _{p} (E)$, such that $M _{0} \cap E _{\infty } =
E$ and $M _{0}E _{\infty } = ME _{\infty }$. This follows from
Galois theory and the projectivity of $\ZZ _{p}$ as a
profinite group (cf. \cite{G}, Theorem~1). In particular, $M
\subseteq M _{0}\Gamma _{n}$, for every sufficiently large index
$n$. We also have $\th (M/E) = \pi _{M _{0}\Gamma _{n}/M} \circ
\th (M _{0}\Gamma _{n}/E)$, which allows us to consider only
the special case of $M = M _{0}\Gamma _{\k }$ and $M _{0}
\neq E$, where $\k $ is chosen so that $[M _{0}\colon E] \mid p ^{\k
}$. Let now $t$ be the rank of $\mathcal{G}(M/E)$, and $E _{1},
\dots , E _{t}$ be cyclic extensions of $E$ in $M$, such that $\prod
_{u=1} ^{t} [E _{u}\colon E] = [M\colon E]$, $E _{1} = \Gamma _{\k
}$ and $E _{2} \dots E _{t} = M _{0}$. Take $E _{1} ^{\prime },
\dots , E _{t} ^{\prime }$ as in Lemma 6.3, denote for convenience
by $\tau _{1}$ the automorphism $\varphi _{\k } (E)$ of $E _{1}$,
and let $\tau _{u}$ be a generator of $\mathcal{G}(E _{u}/E)$, for
every $u \in \{2, \dots , t\}$. Fix an index $x \in I _{p}$, put $b
_{x, \k } (p) = b _{x}$, and identifying $\mathcal{G}(M/E)$ with the
$x$-th component of $\mathcal{G}(M/E) ^{d(p)}$, consider elements $x
_{1}, \dots , x _{t}$ of $E ^{\ast }$ determined so that $\th (M/E)
(x _{u})$ equals the $E _{u} ^{\prime }$-automorphism $\s _{u}$ of
$M$ extending $\tau _{u}$, for each positive integer $u \leq t$. The
assumptions on $\Theta _{p} (E ^{\prime })\colon \ E ^{\prime } \in
\Omega _{p} (E)$, imply that $x _{u} \in N(E _{u} ^{\prime
}/E)\colon \ u = 1, \dots , t$, and $[(E _{1}/E, \tau _{1}, x _{1})]
= b _{x}$. We show as in the proof of Lemma 6.3 (iii) that $[(E
_{u}/E, \tau _{u}, x _{u})] = (p ^{\k }/[E _{u}\colon E])b _{x}$,
for each $u$. Fix an index $u \geq 2$, put $\omega _{u} = p ^{\k
}/[E _{u}\colon E]$, $\s _{1} ^{\prime } = \s _{1}\s _{u}$, $x _{1}
^{\prime } = x _{1}x _{u}$, and denote by $F _{u}$ the fixed field
of the subgroup of $\mathcal{G}(M/E)$ generated by $\s _{1} ^{\prime
}$ and $\s _{u'}\colon \ u ^{\prime } \notin \{1, u\}$. It is easily
verified that $F _{1} \dots F _{t} = M$ and $\prod _{u'=1} ^{t} [F
_{u'}\colon E] = [M\colon E]$, where $F _{u'} = E _{u'}\colon \ u
^{\prime } \neq u$. As ${\rm Ker} (\th (F _{u}/E)) = N(F _{u}/E)$,
$\th (M/E) (x _{1} ^{\prime }) = \s _{1} ^{\prime }$ and $\s _{1}
^{\prime } \in \mathcal{G}(M/F _{u})$, the equality $\th (F/E) = \pi
_{M/E} \circ \th (M/E)$ ensures that $x _{1} ^{\prime } \in N(F
_{u}/E)$. Observe also that $E _{1}E _{u} = E _{1}F _{u}$ and $E
_{u} \cap F _{u} = E$. This implies that $(E _{1}/E, \tau _{1}, x
_{1}) \otimes _{E} (E _{u}/E, \tau _{u}, x _{u})$ is $E$-isomorphic
to $(E _{1}/E, \tau _{1} ^{\prime }, x _{1}x _{u} ^{\omega _{u}})
\otimes _{E} (F _{u}/E, f _{u}, x _{u})$, where $\tau _{1} ^{\prime
} \in \mathcal{G}(E _{1}/E)$ and $f _{u} \in {\mathcal G}(F _{u}/E)$
are induced by $\s _{1} ^{\prime }$ and $\s _{u}$, respectively.
Since $\s _{u} \in \mathcal{G}(M/E _{1})$ and $x _{u} \in N(E
_{1}/E)$, it thereby turns out that $(E _{u}/E, \tau _{u}, x _{u})
\cong (F _{u}/E, f _{u}, x _{u}) \cong (F _{u}/E, f _{u} ^{-1}, x
_{1}) \cong (F _{u}/E, f _{1}, x _{1})$ over $E$, $f _{1}$ being the automorphism of $F _{u}$ induced by $\s _{1}$. Furthermore, if $\Phi
_{u}$ is the extension of $E$ in $E _{1}$ of degree $[E _{u}\colon
E]$, then $E _{u}\Phi _{u} = E _{u}F _{u}$ and, because $x _{1} \in
N(E _{u}/E)$, it follows that $[(E _{u}/E, \tau _{u}, x _{u})] = [(F
_{u}/E, f _{1}, x _{1})] = \omega _{u}[(E _{1}/E, \tau _{1}, x
_{1})] = \omega _{u}b _{x}$, as claimed. The obtained result
indicates that $\th (M/E) (\a _{x}) = (\a _{x}, M/E)$, for each $\a
_{x} \in B(M/E) _{x}$. As $x$ is an arbitrary element of $I _{p}$,
this enables one to complete the proof of Theorem 3.2 by applying
(7.1).

\smallskip
\begin{rema}
Theorem 1.1 and \cite{Ch4}, Theorem~2.1, show that if $(E, v)$ is a
Henselian discrete valued strictly PQL-field, then $\widehat
E(p)/\widehat E$ is a $\ZZ _{p}$-extension, for each $p \in P(E)$.
Therefore, one can take as $E _{\infty }$ the compositum of all $I
\in \Omega (E)$ that are inertial over $E$. Note also that if $E$ is
SQL, then Br$(E)$ is isomorphic to the direct sum $\oplus _{p \in
P(E)} \ZZ (p ^{\infty })$ (by (2.1) (i)), and for each $p \in P(E)$,
the set $\Lambda _{p} (E)$ can be chosen so that one may put $C _{p}
(E) = \{c _{n} (p) = \pi \colon \ n \in \NN \}$, where $\pi $ is a
uniformizer of $(E, v)$. When $E$ is a local field, and for each
$(p, n) \in P(E) \times \NN $, $\varphi _{n} (p)$ is the Frobenius
automorphism of the inertial extension of $E$ in $E _{\rm sep}$ of
degree $p ^{n}$, the sets $H _{p} (E)$, $p \in P(E)$, from the proof
of Theorem 3.2, define the Hase (the norm residue) symbol, in the
sense of \cite{Ko2}, and give rise to the Artin map (cf. \cite{I},
Ch. 6).
\end{rema}

\smallskip
\begin{coro}
In the setting of (2.2) (i), let $M \in \Omega (E)$ and $F$ be a
finite extension of $E$ in $E _{\rm sep}$. Then $N(MF/F) = \{\l \in
F ^{\ast }\colon N _{E} ^{F} (\l ) \in N(M/E)\}$.
\end{coro}

\begin{proof}
It suffices to prove that $N(MF/F)$ includes the preimage of
$N(M/E)$ in $F ^{\ast }$ under $N _{E} ^{F}$. In view of Theorem
3.1, Lemma 2.1, \cite{Ch6}, I, Lemma 4.2 (ii), and of the
PQL-property of $F$, one may consider only the case where $M/F _{0}$
is cyclic, for $F _{0} = M \cap F$. Put $\mu _{0} = N _{F _{0}} ^{F}
(\mu )$, for each $\mu \in F ^{\ast }$. Theorem 3.2 (iii),
\cite{Ch6}, I, Lemma 4.2 (ii), and norm transitivity in towers of
intermediate fields of $MF/E$ imply that if $N _{E} ^{F} (\mu ) \in
N(M/E)$, then $\mu _{0} \in N(M/F _{0})$. At the same time, it
follows from Proposition 2.2 (ii), \cite{Ch6}, I, Corollary 8.5, the
RC-formula and the assumptions on $E$ that Cor$_{F/F _{0}}$ is
injective. Therefore, one deduces from the former part of (4.1)
(iii) (in the general form pointed out in Remark 4.3) that $\mu \in
N(MF/F)$, which proves our assertion.
\end{proof}

\smallskip
Corollary 7.2 generalizes \cite{I}, Theorem 7.6. Applying the
RC-formula, Propositions 2.2 (ii) and 2.3 (i), as well as Lemma 2.1,
norm and corestriction transitivity, and already used known
relations between norms and cyclic algebras, one obtains by the
method of proving Theorem 3.2 the existence of an exact analogue to
the local Hasse symbol in the following situation:

\smallskip
\begin{coro}
Assume that $E$ is a nonreal field, such that every $L \in \Omega
(E)$ is strictly {\rm PQL} with $\rho _{E/L}$ surjective. Then the
maps {\rm Cor}$_{L/E}\colon \ L \in \Omega (E)$, are bijective, and
there are sets  $H(E ^{\prime }) = \{( \ , M ^{\prime }/E ^{\prime
})\colon $ $E ^{\prime \ast } \to \mathcal{G}(M ^{\prime }/E
^{\prime }) ^{{\rm Br}(E')},$ $M ^{\prime } \in \Omega (E ^{\prime
})\}$, $E ^{\prime } \in \Omega (E)$, of group homomorphisms
satisfying the following:

{\rm (i)} $( \ , M ^{\prime }/E ^{\prime })$ is surjective with a
kernel equal to $N(M ^{\prime }/E ^{\prime })$, for each $E ^{\prime
} \in \Omega (E)$, $M ^{\prime } \in \Omega (E ^{\prime })$;

{\rm (ii)} $H _{E'}$ has the properties required by Theorem 1.3
(ii), for every $E ^{\prime } \in \Omega (E)$; furthermore, if $M
\in \Omega (E)$ and $K$ is an intermediate field of $M/E$, then $(\l
, M/K) = (N _{E} ^{K} (\l ), M/E)$, for any $\l \in K ^{\ast }$;

{\rm (iii)} The sets $H(E ^{\prime }), E ^{\prime } \in \Omega (E)$,
are determined by the mappings $( \ , \Gamma /E)$, when $\Gamma $
ranges over finite extensions of $E$ in $E _{\infty }$ of primary
degrees.
\end{coro}

\smallskip
Corollary 7.3 has a partial analogue for a formally real strictly
PQL-field $E$ and the sets $\Omega ^{\prime } (E ^{\prime }) = \{M
^{\prime } \in \Omega (E ^{\prime })\colon \ 2 \dagger [M ^{\prime
}\colon E ^{\prime }]\}$, $E ^{\prime } \in \Omega ^{\prime } (E)$. Specifically, statements (i)$\div $(iii) hold when every $E ^{\prime
} \in \Omega ^{\prime } (E)$ is $p$-quasilocal with Br$(E ^{\prime
}) _{p}$ included in the image of $\rho _{E/E'}$, for each prime $p
> 2$; also, in this case, Cor$_{E'/E}$ induces isomorphisms Br$(E
^{\prime }) _{p} \cong {\rm Br}(E) _{p}$, $p > 2$. This is proved in
the same way as Corollary 7.3.  On the other hand, if $E ^{\prime }
\in \Omega ^{\prime } (E)$ and $E ^{\prime } \neq E$, then $E
^{\prime }$ cannot admit LCFT because it follows from the
Artin-Schreier theory that $E ^{\prime }$ is formally real with at
least $[E^{\prime }\colon E]$ orderings (induced by the action of
$\mathcal{G}(E ^{\prime }/E)$ on $E ^{\prime }$), whence it is not
$2$-quasilocal.

\smallskip
\begin{rema}
Under the hypotheses of Theorem 1.2, let $E _{\rm ab}$ be the
compositum of all $M \in \Omega (E)$, $E _{\rm ab} (p) = E _{\rm ab}
\cap E (p)$, for each $p \in P(E)$, and $N _{1} (E) = \cap _{M \in
\Omega (E)} N(M/E)$. Suppose that $d(p) \in \NN $, for all $p \in
P(E)$, and Pr$_{c} (E ^{\ast })$ is the profinite completion of $E
^{\ast }/N _{1} (E)$. Using Theorem 1.3 and following the proof of
the sufficiency in \cite{CF}, Ch. V, Theorem~1.1, one obtains that
${\rm Pr} _{c} (E ^{\ast })$ is isomorphic to the topological group
product $\prod _{p \in P(E)} \mathcal{G}(E _{\rm ab} (p)/E)
^{d(p)}$, and so generalizes a part of \cite{I}, Proposition~6.3.
\end{rema}

\smallskip
Note finally that every abelian torsion group $T$ admissible by
Proposition 2.2 (ii) is isomorphic to Br$(E(T))$, for some strictly
PQL-field $E(T)$ satisfying the conditions of Corollary 7.3 or its
analogue in the formally real case. This follows from \cite{Ch9},
Corollary~6.6, and (7.2) (when $T$ is divisible, from Remark 2.4 as
well). Hence, all forms of the local reciprocity law and Hasse
symbol admissible by Theorems 1.2, 1.3 and Corollary 7.3 can be
realized.


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\end{document}

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