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.
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References
E. Artin, Questions de base minimale dans la théorie des nombres algébriques, in Collected Papers, Springer, New York, 1982, pp. 229–231.
M. Auslander and D. Buchsbaum, On ramification theory in noetherian rings, American Journal of Mathematics 81 (1959), 749–765.
S. Chase, D. Harrison and A. Rosenberg, Galois theory and cohomology of commutative rings, Memoirs of the American Mathematical Society 52 (1965), 1–19.
L. N. Childs, The group of unramified Kummer extensions of prime degree, Proceedings of the London Mathematical Society 35 (1977), 407–422.
A. Fröhlich, The discriminant of relative extensions and the existence of an integral basis, Mathematika 7 (1960), 15–22.
B. Huppert, Endliche Gruppen I, Springer, New York, 1967.
H. Ichimura, On power integral bases of unramified cyclic extensions of prime degree, Journal of Algebra 235 (2001), 104–112.
H. Ichimura, Hilbert-Speiser fields and the complex conjugation, Journal of the Mathematical Society of Japan 62 (2010), 83–94.
S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, 1970.
H. B. Mann, On integral bases, Proceedings of the American Mathematical Society 9 (1958), 167–173.
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer, New York, 1990.
E. Soverchia, Relative integral bases over a Hilbert class field, Journal of Number Theory 97 (2002), 199–203.
H. Suzuki, A generalization of Hilbert’s theorem 94, Nagoya Mathematical Journal 121 (1991), 161–169.
O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Van Nostrand, New York, 1958.
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Roy, R., Schmid, P. Steinitz classes for Galois extensions of Dedekind rings. Isr. J. Math. 196, 285–293 (2013). https://doi.org/10.1007/s11856-012-0160-7
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DOI: https://doi.org/10.1007/s11856-012-0160-7