Cyclic Galois Extensions and Normal Bases

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.