On the decomposition of cyclic algebras

A cyclic algebra (K/F, σ, a) of degree n has property D(f) if it decomposes as a tensor product of a cyclic algebra of degree e=n/f containing L (the fixed subfield under σ ^e) and a cyclic subalgebra of degree f containing a f-th root of a. Although D(2) holds for every cyclic algebra of degree 4 and exponent 2, D(p) fails for Brauer algebras of degree p ² and exponent p, and D(2) fails for Brauer algebras of degree 8 and exponent 2. Using this, one fills the gap in [6, Theorem 4] and [7, Theorem 7.3.28], to show that the example given there is indeed tensor indecomposable of degree p ² and exponent p. An easy ultraproduct argument provides an example containing all p ^k roots of 1, for all k.