On central division algebras

For any integern such that 8|n or for which there exists an odd primeq such thatq ²|n, there is a central division algebra of dimensionn ² over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X ₁,…,X)_m, the algebra generated over the rationalQ bym(≧2) generic matrices. To the memory of A. A. Albert This paper was originally presented in November, 1971 for publication elsewhere in a volume in honor of Prof. A. A. Albert on the occasion of his 65th birthday. The volume was never published due to the death of Prof. Albert in June 1972.