A note on semidihedral 2-class field towers and $${{\mathbb {Z}}_{2}}$$ Z
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note on semidihedral 2-class field towers and Z2 -extensions Yasushi Mizusawa
Received: 9 September 2013 / Accepted: 12 February 2014 / Published online: 28 June 2014 © Fondation Carl-Herz and Springer International Publishing Switzerland 2014
Abstract We prove that the Galois group of the maximal unramified pro-2-extension over the Z2 -extension of a real quadratic field is never isomorphic to a semidihedral group. Résumé Nous prouvons que le groupe de Galois de la pro-2-extension non ramifiée maximale au-dessus de la Z2 -extension d’un corps quadratique réel n’est jamais isomorphe à un groupe semi-diédral. Keywords
p-Class field towers · Iwasawa theory · Semidihedral groups
Mathematics Subject Classification
11R23
1 Introduction Let p be a prime number. For an algebraic extension k of the rational number field Q, we denote by G(k) the Galois group of the maximal unramified pro- p-extension of k. If the degree [k : Q] is finite, the p-class field tower of k is defined as the sequence of the fixed fields corresponding to the commutator series of G(k). Hence G(k) is often called the Galois group of the p-class field tower. Then G(k) is a Fab pro- p group, i.e., any open subgroup has a finite maximal abelian quotient. Ozaki [10] proved that any finite p-group can be isomorphic to G(k) for some k. However, the theorems of Golod-Shafarevich type (cf. e.g. [8]) imply that if the degree [k : Q] is restricted, then finite p-groups which can be isomorphic to G(k) are also restricted. For example, if p = 2 and k is a quadratic field, no finite 2-groups with at least 6 generators can be isomorphic to G(k) (cf. [8, (10.10.8) Corollary]). On the other hand, various 2-groups with few generators, e.g., the four-group, dihedral groups, generalized quaternion groups and
Y. Mizusawa (B) Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466-8555, Japan e-mail: [email protected]
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semidihedral groups (quasidihedral groups), appear as G(k) for quadratic fields k (cf. [2,4] etc.). Let k∞ be the cyclotomic Z p -extension of a finite extension k of Q, where Z p denotes (the additive group of) the ring of p-adic integers. The Galois group G(k∞ ) can be written as a projective limit of the Galois groups of p-class field towers and has been studied in Iwasawa theory (cf. [9], etc.). Greenberg’s conjecture [3] implies that G(k∞ ) is a Fab pro- p group if k is a totally real number field (cf. [6]). Assuming this conjecture, it seems that the Galois groups G(k∞ ) for totally real k are similar to the Galois groups of p-class field towers. Hence a question arises: Can any finite p-group be isomorphic to G(k∞ ) for some totally real k? We consider this question restricted to the case where p = 2 and k is a real quadratic field. In this case, Mouhib and Movahhedi [7] and the author [5] exhibited a family of fields k such that G(k∞ ) is the four-group, a dihedral group or a generalized quaternion group. The following main theorem, which treats a little more general case, shows
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