International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 7, Pages 375-380
doi:10.1155/S0161171202007767
The Galois extensions induced by idempotents in a Galois algebra
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.
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).