The Galois extensions induced by idempotents in a Galois algebra

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).