Hermite’s theorem via Galois cohomology

  • Matthew Brassil
  • Zinovy ReichsteinEmail author


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.


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

Mathematics Subject Classification

12G05 14G05 


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


  1. 1.
    Artin, M.: Algebra. Prentice Hall Inc., Englewood Cliffs, NJ (1991)zbMATHGoogle Scholar
  2. 2.
    Buhler, J., Reichstein, Z.: On the essential dimension of a finite group. Compos. Math. 106(2), 159–179 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brassil, M., Reichstein, Z.: The Hermite–Joubert problem over \(p\)-closed fields. In: Algebraic Groups: Structure and Actions, Proceedings of Symposium Pure Mathematics, vol. 94, pp. 31–51. American Mathematical Society, Providence, RI (2017)Google Scholar
  4. 4.
    Beauville, A.: Finite subgroups of \({\rm PGL}_2(K)\). In: Vector Bundles and Complex Geometry, Contemporary Mathematics, vol. 522, pp. 23–29. American Mathematical Society, Providence, RI (2010)Google Scholar
  5. 5.
    Berhuy, G., Favi, G.: Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math. 8, 279–330 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Berhuy, G., Favi, G.: Essential dimension of cubics. J. Algebra 278(1), 199–216 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coray, D.F.: Cubic hypersurfaces and a result of Hermite. Duke Math. J. 54, 657–670 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hermite, C.: Sur l’invariant du dix-huitième ordre des formes du cinquième degré. J. Reine Angew. Math. 59, 304–305 (1861)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Joubert, P.: Sur l’equation du sixième degré. C-R. Acad. Sc. Paris 64, 1025–1029 (1867)Google Scholar
  10. 10.
    Kollár, J.: Unirationality of cubic hypersurfaces. J. Inst. Math. Jussieu 1(3), 467–476 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kraft, H.: A result of Hermite and equations of degree 5 and 6. J. Algebra 297, 234–253 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Meyer, A., Reichstein, Z.: The essential dimension of the normalizer of a maximal torus in the projective linear group. Algebra Number Theory 3(4), 467–487 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pfister, A.: Quadratic Forms with Applications to Algebraic Geometry and Topology, London Mathematical Society Lecture Note Series, vol. 217. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  14. 14.
    Reichstein, Z.: Essential dimension. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 162–188. Hindustan Book Agency, New Delhi (2010)Google Scholar
  15. 15.
    Serre, J.-P.: Extensions icosaédriques. In: Seminar on Number Theory, 1979–1980 (French), Exp. 19, 7 pp, Univ. Bordeaux I, Talence (1980)Google Scholar
  16. 16.
    Serre, J.-P.: Galois Cohomology, translated from the French by Patrick Ion and revised by the author. Springer, Berlin (1997)Google Scholar
  17. 17.
    Serre, J.-P.: Cohomological invariants, Witt invariants, and trace forms, notes by Skip Garibaldi. In: Cohomological Invariants in Galois Cohomology, University Lecture Series, vol. 28, pp. 1–100. American Mathematical Society, Providence, RI (2003)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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