International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2002 / Article

Open Access

Volume 29 |Article ID 894789 | 6 pages | https://doi.org/10.1155/S0161171202007767

The Galois extensions induced by idempotents in a Galois algebra

Received07 Jun 2001

Abstract

Let B be a Galois algebra with Galois group G, Jg={bB|bx=g(x)bfor allxB} for each gG, eg the central idempotent such that BJg=Beg, and eK=gK,eg1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)(={gG|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(1eG).

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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