Cyclicity of the unramified Iwasawa module |
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Authors: | Ali Mouhib Abbas Movahhedi |
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Affiliation: | 1. Facult?? Polydisciplinaire de Taza, Universit?? de Fez, B.P 1223, Taza Gare, Morocco 2. Math??matiques et Informatique, XLIM UMR 6172 CNRS, Universit?? de Limoges, 123, Avenue A. Thomas, 87060, Limoges, France
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Abstract: | For a number field k and a prime number p, let k ?? be the cyclotomic Z p -extension of k with finite layers k n . We study the finiteness of the Galois group X ?? over k ?? of the maximal abelian unramified p-extension of k ?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k n . As a consequence, we determine the complete list of real quadratic number fields for which X ?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya. |
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