ON GENUS OF DIVISION ALGEBRAS

Abstract

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D′] ∈ Br (F) , where D′ is a central division F-algebra having the same maximal subfields as D. We show that the fact that quaternion division algebras D and D′ have the same maximal subfields does not imply that the matrix algebras Ml(D) and Ml(D′) have the same maximal subfields for l> 1. Moreover, for any odd n> 1 , we construct a field L such that there are two quaternion division L-algebras D and D′ and a central division L-algebra C of degree and exponent n such that gen(D) = gen(D′) but gen(D⊗ C) ≠ gen(D′⊗ C).