Triality and étale algebras

Part of the Developments in Mathematics book series (DEVM, volume 18)


Trialitarian automorphisms are related to automorphisms of order 3 of the Dynkin diagram of type D 4. Octic étale algebras with trivial discriminant containing quartic subalgebras are classified by Galois cohomology with values in the Weyl group of type D 4. This paper discusses triality for such étale extensions.


Conjugacy Class Weyl Group Isomorphism Class Dynkin Diagram Outer Automorphism 
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We are grateful to Parimala for her unshakable interest in triality, in particular for many discussions at earlier stages of this work and we specially thank Serre for communicating to us his results on Witt and cohomological invariants of the group W(D 4). We also thank Emmanuel Kowalski who introduced us to Magma [2] with much patience, Jean Barge for his help with Galois cohomology and Humphreys and Mühlherr for the reference to the paper [9]. The paper [11] on octic fields was a very useful source of inspiration. Finally, we are highly thankful to the referee for many improvements.


  1. 1.
    L. Beltrametti Über quadratische Erweiterungen étaler Algebren der Dimension vier. Diplomarbeit, Mathematikdepartement, ETH Zürich (2006).
  2. 2.
    W. Bosma, J. Cannon, C. Playoust. The Magma algebra system. I. The user language. J.Symbolic Comput. 24 (3–4), 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Bourbaki. Éléments de mathématique. Groupes and algèbres de Lie. Chapitres 4, 5 et 6. (Masson, Paris, 1981)Google Scholar
  4. 4.
    J.Brillhart. On the Euler and Bernoulli polynomials. J. Reine Angew. Math. 234, 45–64 (1969)MATHMathSciNetGoogle Scholar
  5. 5.
    E.Cartan. Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sci. Math. 49, 361–374 (1925)Google Scholar
  6. 6.
    J.H. Conway, A. Hulpke, J. McKay. On transitive permutation groups. LMS J. Comput. Math. 1 (1998), 1–8MATHMathSciNetGoogle Scholar
  7. 7.
    P.Deligne. Séminaire de géométrie algébrique du Bois-Marie, SGA 4 1/2, Cohomologie étale, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie and J. L. Verdier, Lecture Notes in Mathematics vol. 569. (Springer-Verlag, Berlin, 1977)Google Scholar
  8. 8.
    W.N. Franzsen. Automorphisms of Coxeter Groups. PhD thesis, School of Mathematics and Statistics, University of Sydney (2001).
  9. 9.
    W.N. Franzsen, R.B. Howlett. Automorphisms of nearly finite Coxeter groups. Adv. Geom. 3 (3), 301–338 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Garibaldi, A. A. Merkurjev, J-P. Serre. Cohomological invariants in Galois cohomology. University Lecture Series vol. 28. (American Mathematical Society, Providence, RI, 2003)Google Scholar
  11. 11.
    J.W. Jones, D.P. Roberts. Octic 2-adic fields. J. Number Theory, 128 1410–1429 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M.-A. Knus, A.A. Merkurjev, M.Rost, J.-P. Tignol. The Book of Involutions. American Mathematical Society Colloquium Publications vol. 44. (American Mathematical Society, Providence, R.I., 1998)Google Scholar
  13. 13.
    M.-A. Knus, J.-P. Tignol. Quartic exercises. Inter. J. Math. Math. Sci. 2003 (68), 4263–4323 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M.-A. Knus, J.-P. Tignol. Thin Severi-Brauer varieties. preprint, 2009, arxiv:0912.3359
  15. 15.
    J.-P. Serre. Witt invariants and trace forms. Minicourse, Workshop “From quadratic forms to algebraic groups”, Ascona, organized by Paul Balmer, Eva Bayer and Max-Albert Knus, February 18–23 (2007)Google Scholar
  16. 16.
    J.-P. Serre. Les invariants de W(D 4). Emails, June 5, June 6, June 18, 2009Google Scholar
  17. 17.
    J.Tits. Sur les analogues algébriques des groupes semi-simples complexes. In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, 261–289. (Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957)Google Scholar
  18. 18.
    J.-P. Serre. Sur la trialité et certains groupes qui s’en déduisent. Publ. Math. IHES 2, 14–60 (1959)Google Scholar
  19. 19.
    J.-P. Serre. Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics vol.386. (Springer-Verlag, Berlin, 1974)Google Scholar
  20. 20.
    S.Zweifel. Etale Algebren und Trialität. Diplomarbeit, Mathematikdepartement, ETH Zürich. (2006).

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© Springer New York 2010

Authors and Affiliations

  1. 1.Departement MathematikETH ZentrumZürichSwitzerland
  2. 2.Institut de Mathématique Pure et AppliquéeUniversité catholique de LouvainLouvain-la-NeuveBelgium

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