The Galois extensions induced by idempotents in a Galois algebra

George Szeto,Lianyong Xue

doi:10.1155/s0161171202007767

George Szeto, Lianyong Xue

Open Access

https://doi.org/10.1155/s0161171202007767

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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 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).

Highlights

The Boolean algebra of idempotents for commutative Galois algebras plays an important role
We study the Galois extension BeK where eK = g∈K,eg≠1 eg ∈ Boolean algebra (Ba) for a subgroup K of G
We call B a Galois extension of BG with Galois group G if there exist elements {ai, bi in B, i = 1, 2, . . . , m} for some integer m such that m i=1 ai g(bi) δ1,g for each g

Summary

GEORGE SZETO and LIANYONG XUE

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≠1 eg for a subgroup K of G. 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 given for G(eK ) = N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. A characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1 − eG).

Introduction

Galois algebra with

Galois subgroup

We call a

Bei where each a central

Bei such that each