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Hermite’s theorem via Galois cohomology

  • Matthew Brassil
  • Zinovy ReichsteinEmail author
Article
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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.

Keywords

Hermite’s theorem Quintic polynomial Galois cohomology Tsen-Lang theorem Essential dimension 

Mathematics Subject Classification

12G05 14G05 

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Notes

Acknowledgements

We are grateful to Maxime Bergeron and Rohit Nigpal for helpful comments.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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