The Galois extensions induced by idempotents in a Galois algebra

George Szeto and Lianyong Xue

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 29, 1-6

Abstract:

Let B be a Galois algebra with Galois group G , J g = { b ∈ B | b x = g ( x ) b for all x ∈ B } for each g ∈ G , e g the central idempotent such that B J g = B e g , and e K = ∑ g ∈ K , e g ≠ 1 e g for a subgroup K of G . Then B e K is a Galois extension with the Galois group G ( e K ) ( = { g ∈ G | g ( e K ) = e K } ) containing K and the normalizer N ( K ) of K in G . An equivalence condition is also given for G ( e K ) = N ( K ) , and B e G is shown to be a direct sum of all B e i generated by a minimal idempotent e i . Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension B e G and B ( 1 − e G ) .

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:894789

DOI: 10.1155/S0161171202007767

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