Hermite's theorem via Galois cohomology

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 $F$.