Power-Central Elements in Tensor Products of Symbol Algebras

Abstract

Let A be a central simple algebra over a field F. Let k₁,…, k_r be cyclic extensions of F such that k₁ ⊗_F… ⊗_Fk_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field k_i lies in a symbol F-algebra factor of the same degree as k_i over F. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.

Key Words:

• Armature
• Central simple algebra
• Cohomological dimension
• Symbol algebra
• Valuation

2010 Mathematics Subject Classification:

• 16K20
• 16W60
• 12E15
• 12G10
• 16K50

ACKNOWLEDGMENTS

This article was written while I was an Assistant at Université catholique de Louvain whose hospitality is gratefully acknowledged. The work is a generalization of some results of my Ph.D. thesis. I gracefully thank my advisors A. Quéguiner-Mathieu and J.-P. Tignol for their support, ideas, and for sharing their knowledge. I thank A. S. Merkurjev for providing Corollary 4.5 (a private communication to J.-P. Tignol).