Specialization results in Galois theory

TL;DR

This paper explores specialization in Galois theory, providing new results on Hilbert-Grunwald type theorems, describing separable closures of PAC fields, and analyzing fields of definition of covers using Hurwitz spaces.

Contribution

It introduces novel applications of specialization results in Galois theory, including explicit descriptions of separable closures and criteria for fields of definition of covers.

Findings

Infinitely many specializations with prescribed local degrees exist for certain covers.

Separable closure of a PAC field is generated by elements with specific polynomial properties.

A criterion for an étale algebra to be a specialization of a cover is established.

Abstract

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an \'etale algebra $\prod_lE_l/k$ over a field $k$ to be the specialization of a $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$. The question is reduced to finding $k$-rational points on a certain $k$-variety, and then studied over the various fields $k$ of our applications.