The Galois extensions induced by idempotents in a Galois algebra

Szeto, George; Xue, Lianyong

doi:https://doi.org/10.1155/S0161171202007767

International Journal of Mathematics and Mathematical Sciences

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Volume 29 |Article ID 894789 | https://doi.org/10.1155/S0161171202007767

George Szeto, Lianyong Xue, "The Galois extensions induced by idempotents in a Galois algebra", International Journal of Mathematics and Mathematical Sciences, vol. 29, Article ID 894789, 6 pages, 2002. https://doi.org/10.1155/S0161171202007767

The Galois extensions induced by idempotents in a Galois algebra

George Szeto¹ and Lianyong Xue¹

¹Department of Mathematics, Bradley University, USA

Received07 Jun 2001

Abstract

Let B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(x)b for all x∈B} for each g∈G, eg the central idempotent such that BJg=Beg, and eK=∑g∈K,eg≠1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)(={g∈G|g(eK)=eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK)=N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1−eG).

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Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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