The Galois extensions induced by idempotents in a Galois algebra

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 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).