Hermite's theorem via Galois cohomology

TL;DR

This paper provides a new proof of Hermite's 1861 theorem on degree 5 field extensions, utilizing Galois cohomology techniques under mild assumptions on the base field.

Contribution

The paper introduces a novel Galois cohomology-based proof of Hermite's theorem, expanding the understanding of polynomial forms in degree 5 extensions.

Findings

New proof of Hermite's theorem using Galois cohomology

Conditions on the base field for the proof to hold

Insight into polynomial minimal forms in degree 5 extensions

Abstract

An 1861 theorem of Hermite asserts that for every field extension $E/F$ of degree $5$ there exists an element of $E$ whose minimal polynomial over $F$ is of the form $f(x) = x^5 + c_2 x^3 + c_4 x + c_5$ for some $c_2, c_4, c_5 \in F$. We give a new proof of this theorem using techniques of Galois cohomology, under a mild assumption on $F$.