A Kummer theory is presented which does not need roots of unity in the ground ring. For R commutative with p~ e R we study the group of cyclic Galois extensions of fixed degree p" in detail. Our theory is well suited for dealing with cyclic p"-extensions of a number field K which are unramified outside p. We then consider the group Gal(tfK[p~ ], C' n) of all such extensions, and its subgroup NB^tp-], C«) of extensions with integral normal basis outside p . For the size of the latter we get a

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... ple asymptotic formula (n -► oo), and the discrepancy between the two groups is in some way measured by the defect ô in Leopoldt's conjecture.