Splitting fields of central simple algebras of exponent two. (English) Zbl 1415.12005
Summary: By Merkurjev’s Theorem every central simple algebra of exponent two is Brauer equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over a given field is split, then there exists no central simple algebra of exponent two over this field. This note provides an independent elementary proof for the latter fact.
MSC:
| 12G05 | Galois cohomology |
| 11E04 | Quadratic forms over general fields |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
| 16K50 | Brauer groups (algebraic aspects) |
References:
| [1] | Gille, P.; Szamuely, T., Central Simple Algebras and Galois Cohomology (2006), Cambridge University Press · Zbl 1137.12001 |
| [2] | Knus, M.-A.; Merkurjev, A. S.; Rost, M.; Tignol, J.-P., The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44 (1998), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0955.16001 |
| [3] | Merkurjev, A. S., On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Sov. Math. Dokl., 24, 546-551 (1981), (in Russian); English translation: · Zbl 0496.16020 |
| [4] | Rowen, L. H., Division algebras of exponent 2 and characteristic 2, J. Algebra, 90, 71-83 (1984) · Zbl 0548.16020 |
| [5] | Rowen, L. H., Ring Theory, vol. II, Pure and Applied Mathematics, vol. 128 (1988), Academic Press, Inc.: Academic Press, Inc. Boston, MA · Zbl 0651.16001 |
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