International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 7, Pages 375-380

The Galois extensions induced by idempotents in a Galois algebra

George Szeto and Lianyong Xue

Department of Mathematics, Bradley University, Peoria 61625, IL, USA

Received 7 June 2001

Copyright © 2002 George Szeto and Lianyong Xue. 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.


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).