Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A propos d'une version faible du probl\`eme inverse de Galois   This paper deals with the Weak Inverse Galois Problem which, for a given
field $k$, states that, for every finite group $G$, there exists a finite
separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to
explain how one can generically produce families of fields which fulfill this
problem, but which do not fulfill the usual Inverse Galois Problem. We show
that this holds for, e.g., the fields $\mathbb{Q}^{{{\rm{sol}}}}$,
$\mathbb{Q}^{{{\rm{tr}}}}$, $\mathbb{Q}^{{{\rm{pyth}}}}$, and for the maximal
pro-$p$-extensions of $\mathbb{Q}$. Moreover, we show that, for every finite
non-trivial group $G$, there exists many fields fulfilling the Weak Inverse
Galois Problem, but over which $G$ does not occur as a Galois group. As a
further application, we show that every field fulfills the regular version of
the Weak Inverse Galois Problem.

14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex