On the decomposition of cyclic algebras

Rowen, Louis H.;Tignol, Jean-Pierre
(1996) Israel Journal of Mathematics — Vol. 96, p. 553-578 (1996)

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Abstract
A cyclic algebra (K/F, s, 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 s^e) and a cyclic subalgebra of degree f containing an 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.
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Rowen, L. H., & Tignol, J.-P. (1996). On the decomposition of cyclic algebras. Israel Journal of Mathematics, 96, 553-578. https://doi.org/10.1007/BF02937323 (Original work published 1996)