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Galois Extensions as Modules over the Group Ring

Published online by Cambridge University Press:  20 November 2018

Gerald Garfinkel  and
Morris Orzech
Gerald Garfinkel
Affiliation:
University of Illinois, Urbana, Illinois
Morris Orzech
Affiliation:
University of Illinois, Urbana, Illinois and Queen's University, Kingston, Ontario
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Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 242 - 248
Copyright
Copyright © Canadian Mathematical Society 1970

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