On the maximal unramified pro-2-extension of certain cyclotomic $$\mathbb {Z}_2$$ Z 2 -extensions
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On the maximal unramified pro-2-extension of certain cyclotomic Z2 -extensions Abdelmalek Azizi1 · Mohammed Rezzougui1 · Abdelkader Zekhnini2 Accepted: 4 May 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized G ab is of type (2, 2m ), with m ≥ 2, to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified pro-2-extension of the cyclotomic Z2 -extension of certain number fields. Illustration is given by some real quadratic fields. Keywords Iwasawa theory · Z2 -extension · 2-Class field tower · Real quadratic field · 2-Class group · Metacyclic and non-metacyclic 2-group Mathematics Subject Classification Primary 11R23 · 20D15 · Secondary 11R11 · 11R20 · 11R29 · 11R32 · 11R37
1 Introduction Let p be a prime integer and Z p be the ring of all p-adic integers. A Galois extension K of a field k is called a Z p -extension over k if the Galois group Gal(K /k) is topologically isomorphic to the additive group Z p . Every number field k has at least one Z p -extension, namely the cyclotomic Z p -extension k∞ obtained by the compositum k∞ = kQ∞ , where Q∞ is the cyclotomic Z p -extension of the field of rational numbers Q. For each positive integer n, denote by kn the unique intermediate field of k∞ /k of degree p n over k, and by A(kn ) the p-Sylow subgroup of the ideal class group of kn . Then the Iwasawa module
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Mohammed Rezzougui [email protected] Abdelmalek Azizi [email protected] Abdelkader Zekhnini [email protected]
1
Mathematics Department, Sciences Faculty, Mohammed First University, Oujda, Morocco
2
Department of Mathematics and Informatics, Pluridisciplinary faculty, Mohammed First University, Nador, Morocco
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A. Azizi et al.
X = X (k∞ ) is defined as the projective limit of the family (A(kn ))n with respect to the norm n mapping. Let λ, μ and ν be the Iwasawa invariants, then the order of A(kn ) is p λn+μ p +ν for n sufficiently large. The Greenberg’s conjecture [9] states that if k is totally real then λ = μ = 0, i.e. X is finite. This conjecture is generally still open, except for some particular cases. Let k∞ be the cyclotomic Z2 -extension of a number field k. Denote by L(k∞ ) (resp. L(kn )) the maximal unramified pro-2-extension of k∞ (resp. n-th layer kn of k∞ /k), and by L(k∞ ) the maximal abelian subextension of L(k∞ )/k∞ . The Galois group G = Gal(L(k∞ )/k∞ ) is isomorphic to the inverse limit of the Galois groups Gal(L(kn )/kn ) with respect to the restriction maps, and it is well known that the maximal abelian quotient group G ab satisfies G ab Gal(L(k∞ )/k∞ ) X (k∞ ).
In this paper, we begin by stating a necessary and sufficient criterion for a finite metabelian 2-group whose abelianized is of type (2, 2m ), with m ≥ 2, to be metacyclic or not, by developing results on the rank of
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