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Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

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Becher, Karim Johannes [Universiteit Antwerpen] Grenier-Boley, Nicolas [Université de Rouen Normandie] Tignol, Jean-Pierre [UCL]

Two different proofs are given showing that a quaternion algebra Q defined over a quadratic ́etale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2018
Language Anglais
Journal information "Archiv der Mathematik" - Vol. 111, no.2, p. 135-143 (2018)
Peer reviewed yes
Publisher Springer International Publishing AG
issn 0003-889X
e-issn 1420-8938
Publication status Publié
Affiliation UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique
Links
  1. Becher Karim Johannes, Grenier-Boley Nicolas, Tignol Jean-Pierre, Involutions and stable subalgebras, 10.1016/j.jalgebra.2017.09.026
  2. P. Draxl, Über gemeinsame separabel-quadratische Zerfällungskörper von Quaternionenalgebren, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1975, 251–259.
  3. R. Elman, N. Karpenko, and A. Merkurjev, The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications 56, American Mathematical Society, Providence, RI, 2008.
  4. Gille Philippe, Szamuely Tamas, Central Simple Algebras and Galois Cohomology, ISBN:9780511607219, 10.1017/cbo9780511607219
  5. M.-A. Knus, Sur la forme d’Albert et le produit tensoriel de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), 333–337.
  6. M.-A. Knus, A. S. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications 44, American Mathematical Society, Providence, RI, 1998.
  7. J. Tits, Sur les produits tensoriels de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), 329–331.
Bibliographic reference Becher, Karim Johannes ; Grenier-Boley, Nicolas ; Tignol, Jean-Pierre. Transfer of quadratic forms and of quaternion algebras over quadratic field extensions. In: Archiv der Mathematik, Vol. 111, no.2, p. 135-143 (2018)
Permanent URL http://hdl.handle.net/2078.1/201050