Automorphisms of congruence function fields

Abstract

Let k be a finite field. For a function field K over k and m > 3, it is proven that there are infinitely many non-isomorphic function fields L such that L/K is a separable extension of degree m and Aut£ L -{Id}. It is also shown that for a finite group G, there are infinitely many non-isomorphic function fields L/k such that Aut* L = G. Finally, given any finite nilpotent group G such that |G| > 1 and (|<7|, \k\ -1) = 1 and any function field K over k 9 there are infinitely many non-isomorphic function fields L over k with Gal(L/K) = Aut* L £* G.

Finite Groups as Galois Groups of Function Fields With Infinite Field of Constants

“…Rzedowski and Villa [7] proved an analogue of Stichtenoth's result for congruence function fields without restriction on the genus, provided that in the extension E/k(x) there exist prime divisors of degree one, one ramified and another unramified. In [1] we remove the ramification restrictions given in [7].…”

“…https://doi.org/10.1017/S1446788710000108 [7] Finite groups as Galois groups of function fields with infinite field of constants 307 …”

“…Given a function field K /k, we first choose a suitable y ∈ K and construct a suitable C-improvement E 1 /k(y) of E/k(x) (see [7]) such that the field of constants…”

“…The main goal of this paper is to establish an analogue of the main result of [7] for an infinite field of constants k now under one ramification restriction. We prove that given any infinite field k, if E/k(x) is any separable extension, where k is the full field of constants of E, such that a place of k(x) of degree one is ramified in E, then for any function field K /k, there exist infinitely many nonisomorphic function fields over k such that L/K is a separable extension of the same degree as E/k(x) and Aut K L = Aut k L ∼ = Aut k(x) E. This is Theorem 4.3.…”

“…Given a function field K /k, we first choose a suitable y ∈ K and construct a suitable C-improvement E 1 /k(y) of E/k(x) (see [7]) such that the field of constants of L = E 1 K is k, while [L : K ] = [K : k(y)] and the intermediate fields of L other than K have large enough genus (Proposition 3.5). With this condition, it follows that any element of the group of automorphisms of L over k restricts to an automorphism of K (Proposition 3.6).…”