Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complements on Furtw\"angler's second theorem and Vandiver' s cyclotomic
  integers   This article deals with a conjecture generalizing the second case of Fermat's
Last Theorem, called $SFLT2$ conjecture: {\it Let $p>3$ be a prime,
$K:=\Q(\zeta)$ the $p$th cyclotomic field and $\Z_K$ its ring of integers. The
diophantine equation $(u+v\zeta)\Z_K=\mk w_1^p$, with $u,v\in\Z\backslash\{0\}$
coprime, $uv\equiv 0 \bmod p$ and $\mk w_1$ ideal of $\Z_K$, has no solution.}
Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing
$uv$, $n$ the order of $\frac{v}{u}\bmod q$, $\xi$ a primitive $n$th root of
unity and $M:=\Q(\xi,\zeta)$. The aim of this complement of the article [GQ] of
G. Gras and R. Qu\^eme on the same topic, is to exhibit some strong properties
of the decomposition of the primes $\mk Q$ of $\Z_M$ over $q$ in certain Kummer
$p$-extensions of the field $M$, to derive from them a weak conjecture which
implies that the SFLT2 equation can always take the reduced form $u+\zeta v\in
K^{\times p}$ and to set a conjecture implying SFLT2.

35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
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The categories of this paper are: supr-con
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci