, Volume 196, Issue 1, pp 285-293
Date: 19 Oct 2013

Steinitz classes for Galois extensions of Dedekind rings

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Abstract

Galois extensions of commutative rings have been studied by Chase, Harrison, Rosenberg and others. Suppose B|A is such an extension with (finite) group G where both A and B are Dedekind rings. Then the Steinitz class s B|A is an element in the class group Cl(A) which vanishes if and only if B is a free A-module. It is shown that s B|A = 1 except possibly when the characteristic char(A) ≠ 2 and G has a cyclic Sylow 2-subgroup ≠ 1. In the exceptional case there is a unique (normal) subgroup H of G with index 2 and s B|A = s C|A where C = B H is the fixed ring. The remaining quadratic case is known and easily treated.