Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cyclic Galois extensions and normal bases

Author: C. Greither
Journal: Trans. Amer. Math. Soc. 326 (1991), 307-343
MSC: Primary 11R23; Secondary 11R32, 13B05
MathSciNet review: 1014248
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Abstract: A Kummer theory is presented which does not need roots of unity in the ground ring. For $ R$ commutative with $ {p^{ - 1}} \in R$ we study the group of cyclic Galois extensions of fixed degree $ {p^n}$ in detail. Our theory is well suited for dealing with cyclic $ {p^n}$-extensions of a number field $ K$ which are unramified outside $ p$. We then consider the group $ \operatorname{Gal}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of all such extensions, and its subgroup $ {\text{NB}}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of extensions with integral normal basis outside $ p$. For the size of the latter we get a simple asymptotic formula $ (n \to \infty)$, and the discrepancy between the two groups is in some way measured by the defect $ \delta $ in Leopoldt's conjecture.

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Keywords: Galois extensions of rings, descent, integral normal bases, Leopoldt's conjecture
Article copyright: © Copyright 1991 American Mathematical Society