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k Tensor product of algebras
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A/B is Galois A/B is left Accordingly algebra of finite Amer arbitrary element Artinian ring B-basis B.V-A-irreducible bounded degree central simple algebra coincides completely reducible consequence of Prop contraction map conversely Corollary dimensional cyclic extension division algebra division ring evidently exists an element F-group f-regular intermediate ring finite Galois finite rank Galois and finite Galois extensions Galois group Galois theory h-Galois and locally Hence homomorphism idempotent infinite inner Galois intermediate field invariant irreducible isomorphic Kummer extension left algebraic left locally finite left q-system Lemma Let A/B linearly disjoint locally Galois Math matrix units Nagahara necessary and sufficient non-zero element Okayama Univ outer Galois Proposition q-Galois and left q-system regular subring resp right Artinian ring of A/B simple intermediate ring simple ring subgroup subset F suffices to prove system of matrix Theorem Tominaga topological unital simple subring unital subring w-q-Galois whence it follows