Cohomological operations for the Brauer group and Witt kernels of a quartic field extension. (English) Zbl 1426.11021
Summary: Let \(F\) be a field, char \(F\neq 2\), \(\sqrt{-1}\in F^\ast\), \(L/F\) a quartic field extension. We investigate the divided power operation \(\gamma_2\) on the group \(_2\mathrm{Br}(L/F)\). In particular, we show that any element of \(\gamma_2(L/F)\) is a symbol \(\varphi \otimes \langle \langle a,\, b^2-d \rangle \rangle\), where \(a,b\in F\), \(d=\operatorname{disc}(L/F)\), and \(\varphi \simeq Tr_{L/F}(t^2)\) is the quadratic trace form associated with the extension \(L/F\). As an application, we obtain certain results on the Stifel-Whitney maps \(w_n:W(L/F)\to k_n(F)\).
MSC:
| 11E04 | Quadratic forms over general fields |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
References:
| [1] | Arason, J. K. and Pfister, A., Beweis des Krullschen Durchschnittsatzes für den WittringInvent. Math.12 (1971) 173-176. · Zbl 0212.37302 |
| [2] | Elman, R., Karpenko, N. A. and Merkurjev, A. S., The algebraic and geometric theory of quadratic forms (American Mathematical Society, 2008). · Zbl 1165.11042 |
| [3] | Fitzgerald, R. W., Witt kernels of function field extensions, Pacific J. Math.109(1) (1983) 89-106. · Zbl 0486.10016 |
| [4] | Kahn, B., Comparison of some field invariants, J. Algebra220(2) (2000) 485-492. · Zbl 0972.11018 |
| [5] | Kahn, B., Formes Quadratiques sur un Corps (American Mathematical Society, 2009). · Zbl 1191.11002 |
| [6] | Lam, T. Y., Leep, D. B. and Tignol, J. P., Biquaternion algebras and quartic extensions, Publ. Math. IHES.77 (1993) 63-102. · Zbl 0801.11048 |
| [7] | Merkurjev, A. S., On the norm residue symbol of degree two, Sovrem. Math. Dokl.24 (1981) 546-551. · Zbl 0496.16020 |
| [8] | Rost, M., Serre, J.-P. and Tignol, J.-P., La forme trace d’une algebre simple centrale de degre \(4\), C. R. Acad. Sci. Paris, Ser. I342(2) (2006) 83-87. · Zbl 1110.16014 |
| [9] | Sivatski, A. S., The Witt ring kernel of a fourth degree field extension and related problems, J. Pure Appl. Algebra214(1) (2010) 61-70. · Zbl 1226.11050 |
| [10] | Sivatski, A. S., Central simple algebras of prime exponent and divided power operations, J. K-theory11(1) (2013) 113-123. · Zbl 1275.16020 |
| [11] | D. B. Shapiro, Similarities, quadratic forms and Clifford algebras, thesis, University of California, Berkeley, California (1974). |
| [12] | Springer, T. A., Sur les formes quadratiques d’indice zéro, C. R. Acad. Sci.234 (1952) 1517-1519. · Zbl 0046.24303 |
| [13] | Vial, C., Operations in Milnor \(K\)-theory, J. Pure Appl. Algebra213(7) (2009) 1325-1345 · Zbl 1185.19002 |
| [14] | Wadsworth, A. R., Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc.208 (1975) 352-358. · Zbl 0336.15013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
