Kummer subspaces of tensor products of cyclic algebras

Author:
Adam Chapman

Journal:
Proc. Amer. Math. Soc. **145** (2017), 4247-4253

MSC (2010):
Primary 16K20; Secondary 05C38, 16W60

DOI:
https://doi.org/10.1090/proc/13599

Published electronically:
April 6, 2017

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

- [CGM16]
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** - [Cha09]
Adam Chapman,
*Polynomial equations over division rings*, Thesis (M.Sc.), Bar-Ilan University, 2009. - [Cha13]
-,
*-central subspaces of central simple algebras*, Thesis (Ph.D.), Bar-Ilan University, 2013. - [CV12]
Adam Chapman and Uzi Vishne,
*Clifford algebras of binary homogeneous forms*, J. Algebra**366**(2012), 94-111. MR**2942645** - [CV13]
-,
*Square-central elements and standard generators for biquaternion algebras*, Israel J. Math.**197**(2013), no. 1, 409-423. MR**3096621** - [Lam73]
T. Y. Lam,
*The algebraic theory of quadratic forms*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR**0396410 (53 #277)** - [Mat16]
Eliyahu Matzri,
*Symbol length in the Brauer group of a field*, Trans. Amer. Math. Soc.**368**(2016), no. 1, 413-427. MR**3413868** - [MS82]
A. S. Merkurev and A. A. Suslin,
*-cohomology of Severi-Brauer varieties and the norm residue homomorphism*, Izv. Akad. Nauk SSSR Ser. Mat.**46**(1982), no. 5, 1011-1046, 1135-1136. MR**675529 (84i:12007)** - [MV12]
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** - [Rac09]
Mélanie Raczek,
*On ternary cubic forms that determine central simple algebras of degree 3*, J. Algebra**322**(2009), no. 5, 1803-1818. MR**2543635 (2010h:16043)** - [Rev77]
Ph. Revoy,
*Algèbres de Clifford et algèbres extérieures*, J. Algebra**46**(1977), no. 1, 268-277. MR**0472881 (57 #12568)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
16K20,
05C38,
16W60

Retrieve articles in all journals with MSC (2010): 16K20, 05C38, 16W60

Additional Information

**Adam Chapman**

Affiliation:
Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208 Israel

Email:
adam1chapman@yahoo.com

DOI:
https://doi.org/10.1090/proc/13599

Keywords:
Central simple algebras,
cyclic algebras,
Kummer spaces,
generic algebras,
graphs

Received by editor(s):
May 5, 2014

Received by editor(s) in revised form:
October 18, 2016, and November 22, 2016

Published electronically:
April 6, 2017

Communicated by:
Jerzy Weyman

Article copyright:
© Copyright 2017
American Mathematical Society