Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the realization of subgroups of $PGL(2,F)$, and their automorphism
  groups, as Galois groups over function fields   Let $F$ be any field. We give a short and elementary proof that any finite
subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field
$F(x)$. We also develop a theory of descent to subfields of $F$. This enables
us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as
Galois groups.

25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int