## 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

Zbl 0826.11052
Full Text:

### References:

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