On the decomposition of cyclic algebras

A cyclic algebra (K/F, σ, a) of degreen hasproperty D(f) if it decomposes as a tensor product of a cyclic algebra of degreee=n/f containingL (the fixed subfield underσ ^e) and a cyclic subalgebra of degreef containing af-th root ofa. AlthoughD(2) holds for every cyclic algebra of degree 4 and exponent 2,D(p) fails for Brauer algebras of degreep ² and exponentp, andD(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 degreep ² and exponentp. An easy ultraproduct argument provides an example containing allp ^k roots of 1, for allk.