Cyclic Galois extensions and normal bases

Greither, C.

doi:10.1090/S0002-9947-1991-1014248-8

Cyclic Galois extensions and normal bases
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Trans. Amer. Math. Soc. 326 (1991), 307-343 Request permission

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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