Hilbert specialization results with local conditions

TL;DR

This paper develops methods to construct specializations of finite extensions of rational function fields over characteristic zero fields, achieving prescribed local behaviors and unifying ramified and unramified cases.

Contribution

It provides a non-Galois analog of previous ramified results and unifies ramified and unramified specialization techniques over number fields.

Findings

Constructed specializations with prescribed local conditions.

Extended results to non-Galois and ramified cases.

Unified ramified and unramified specialization frameworks.

Abstract

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with specified local behavior at any given finite set of primes of $k$. First, we give a full non-Galois analog of a result with a ramified type conclusion from a preceding paper and next we prove a unifying statement which combines our results and previous work devoted to the unramified part of the problem in the case $k$ is a number field.