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Imaginary quadratic fields \(k\) with cyclic \(\text{Cl}_2(k^1)\). (English) Zbl 0919.11074
Let \(k\) be an imaginary quadratic number field, and \(Cl_2(k)\) denote the 2-class group of \(k\). Let \(k^1\) denote the Hilbert 2-class field of \(k\), and \(k^2=(k^1)^1\). If \(G =\text{Gal}(k^2/k)\), then the commutator subgroup \(G'= \text{Gal}(k^2/k^1)\simeq Cl_2(k^1)\), and \(G^{ab}= G/G'\simeq \text{Gal} (k^1/k) \simeq Cl_2(k)\). Moreover it is known that if \(G/G'\simeq (2,4)\) then rank \(Cl_2(k^1)\leq 3\), and further if rank \(Cl_2(k^1)<3\) then the 2-class field tower of \(k\) terminates at \(k^2\). The authors are interested in searching for fields with finite 2-class field tower of length at least 3. With this motivation they study the 2-rank of \(Cl_2(k^1)\), and they determine the imaginary quadratic fields \(k\) such that \(Cl_2(k^1)\) are cyclic.
Since one of the authors [F. Lemmermeyer, J. Théor. Nombres Bordx. 6, 261-272 (1994; Zbl 0826.11052)] determined the cases where \(\text{Gal} (k^2/k)\) are abelian or metacyclic, in this paper they treat the non-metacyclic case. For this the authors first prove that if \(Cl_2(k^1)\) is cyclic, then \(Cl_2(k)\subseteq (2,2^m)\) using Schur multipliers. Next the group structure of non-metacyclic 2-group \(G\) such that \(G^{ab}\simeq(2,2^m)\) with \(m>1\) is studied in detail. Using this result they finally characterize the imaginary quadratic fields \(k\) such that \(Cl_2(k^1)\) are cyclic in terms of quadratic and quartic symbols about the ramified primes in \(k\), and determine the group structure of \(\text{Gal} (k^2/k)\).

MSC:
11R37 Class field theory
11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
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