Hermite's theorem via Galois cohomology

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 $F$.

6 pages