On the decomposition of cyclic algebras |
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Authors: | L H Rowen J -P Tignol |
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Institution: | 1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel 2. Département de Mathématiques, Université catholique de Louvain, B-1348, Louvain-la-Neuve, Belgium
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Abstract: | 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. |
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