Kummer subspaces of tensor products of cyclic algebras

Chapman, Adam

doi:10.1090/proc/13599

Kummer subspaces of tensor products of cyclic algebras
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Proc. Amer. Math. Soc. 145 (2017), 4247-4253 Request permission

Abstract:

We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal Kummer spaces in any tensor product of cyclic algebras of prime degree, classify all the monomial Kummer spaces in tensor products of cyclic algebras of degree 3, and provide an upper bound for the dimension of Kummer spaces in the generic tensor product of cyclic algebras of degree 3.

References

Adam Chapman, David J. Grynkiewicz, Eliyahu Matzri, Louis H. Rowen, and Uzi Vishne, Kummer spaces in symbol algebras of prime degree, J. Pure Appl. Algebra 220 (2016), no. 10, 3363–3371. MR 3497965, DOI 10.1016/j.jpaa.2016.04.003
Adam Chapman, Polynomial equations over division rings, Thesis (M.Sc.), Bar-Ilan University, 2009.
—, $p$-central subspaces of central simple algebras, Thesis (Ph.D.), Bar-Ilan University, 2013.
Adam Chapman and Uzi Vishne, Clifford algebras of binary homogeneous forms, J. Algebra 366 (2012), 94–111. MR 2942645, DOI 10.1016/j.jalgebra.2012.05.012
Adam Chapman and Uzi Vishne, Square-central elements and standard generators for biquaternion algebras, Israel J. Math. 197 (2013), no. 1, 409–423. MR 3096621, DOI 10.1007/s11856-013-0005-z
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
Eliyahu Matzri, Symbol length in the Brauer group of a field, Trans. Amer. Math. Soc. 368 (2016), no. 1, 413–427. MR 3413868, DOI 10.1090/tran/6326
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
Eliyahu Matzri and Uzi Vishne, Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3, Manuscripta Math. 137 (2012), no. 3-4, 497–523. MR 2875290, DOI 10.1007/s00229-011-0482-5
Mélanie Raczek, On ternary cubic forms that determine central simple algebras of degree 3, J. Algebra 322 (2009), no. 5, 1803–1818. MR 2543635, DOI 10.1016/j.jalgebra.2009.05.039
Ph. Revoy, Algèbres de Clifford et algèbres extérieures, J. Algebra 46 (1977), no. 1, 268–277 (French). MR 472881, DOI 10.1016/0021-8693(77)90404-5