Identifying Frobenius elements in Galois groups

TL;DR

This paper introduces a new method for identifying Frobenius elements in Galois groups of global fields, extending classical criteria and aiding in understanding Galois group conjugacy classes.

Contribution

It generalizes Euler's criterion to arbitrary Galois extensions, providing a novel approach for comparing splitting fields and identifying conjugacy classes.

Findings

Method successfully determines Frobenius elements in various Galois extensions

Extends classical Euler's criterion to broader contexts

Facilitates comparison of splitting fields and Galois group analysis

Abstract

We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler's criterion. It is a part of the general question how to compare splitting fields and identify conjugacy classes in Galois groups, that we will discuss as well.