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 R commutative with p − 1 ∈ R {p^{ - 1}} \in R we study the group of cyclic Galois extensions of fixed degree p n {p^n} in detail. Our theory is well suited for dealing with cyclic p n {p^n} -extensions of a number field K K which are unramified outside p p . We then consider the group Gal ⁡ ( O K [ p − 1 ] , C p n ) \operatorname {Gal}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}}) of all such extensions, and its subgroup NB ( O K [ p − 1 ] , C p n ) {\text {NB}}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}}) of extensions with integral normal basis outside p p . For the size of the latter we get a simple asymptotic formula ( n → ∞ ) (n \to \infty ) , and the discrepancy between the two groups is in some way measured by the defect δ \delta in Leopoldt’s conjecture.