Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic extensions in free groups   The aim of this paper is to unify the points of view of three recent and
independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich
and Miasnikov 2002), where similar modern versions of a 1951 theorem of
Takahasi were given. We develop a theory of algebraic extensions for free
groups, highlighting the analogies and differences with respect to the
corresponding classical field-theoretic notions, and we discuss in detail the
notion of algebraic closure. We apply that theory to the study and the
computation of certain algebraic properties of subgroups (e.g. being malnormal,
pure, inert or compressed, being closed in certain profinite topologies) and
the corresponding closure operators. We also analyze the closure of a subgroup
under the addition of solutions of certain sets of equations.

36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph