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On Demuškin groups with involution. (English) Zbl 0715.11064

The main result of the paper shows a certain Galois group \(G= \mathrm{Gal}(\tilde k_{S_ p}/k_{\infty})\) to be a Demushkin group. Here \(k\) is an algebraic number field of CM-type containing an \(p\)-th root of unity, \(p\ne 2\), and \(k_{\infty}\) is the cyclotomic \(\mathbb Z_p\)-extension of \(k\) for which the Iwasawa \(\mu\)-invariant is required to vanish. Finally, \(\tilde k_{S_p}\) is the maximal \(p\)-extension of \(k\) that is unramified outside the set \(S_p\) of primes of \(k\) above \(p\) and that is positively ramified at \(p\) (we omit the precise definition). It is shown that \(G\) has rank \(2g\) with \(g\) equal to the Iwasawa \(\lambda\)-invariant of the minus part of the maximal abelian unramified \(p\)-extension of \(k_{\infty}\); moreover, the existence of a particular generating set satisfying one explicitly given defining relation is proved.

MSC:

11R32 Galois theory
11R34 Galois cohomology
11R23 Iwasawa theory
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References:

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