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.
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We are grateful to Maxime Bergeron and Rohit Nigpal for helpful comments.
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Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery Grant 253424-2017.
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Brassil, M., Reichstein, Z. Hermite’s theorem via Galois cohomology. Arch. Math. 112, 467–473 (2019). https://doi.org/10.1007/s00013-019-01299-5
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DOI: https://doi.org/10.1007/s00013-019-01299-5