Geodesic
Galois Extensions as Modules over the Group Ring
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 242-248

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Garfinkel, Gerald; Orzech, Morris. Galois Extensions as Modules over the Group Ring. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 242-248. doi: 10.4153/CJM-1970-031-6
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