Abstract
Suppose, K is a number field, and p is an odd prime. Setting R K = D K [ p −1 ] with D K being the ring of integers of K, the isomorphism classes of Z p -extensions of R K form a Z p -module H 1 ( R K , Z p ). Leopoldt's conjecture states that H 1 ( R K , Z p ) ≅ Z p r 2 + 1 . In this paper we define a Z p -submodule H( R K , Z p ) of H 1 ( R K , Z p ) consisting of those classes of Z p -extensions having a normal basis over R K . We prove for a complex multiplication field K that there is an isomorphism H( R K , Z p ) ≅ Z p r 2 + 1 .
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