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Imaginary quadratic fields \(k\) with \(\text{Cl}_ 2(k)\simeq(2,2^ m)\) and rank \(\text{Cl}_ 2(k^ 1)=2\). (English) Zbl 1063.11038

The imaginary quadratic fields \(k\) such that both \(k\) and its Hilbert 2-class field have 2-class groups of rank 2 are characterized in this article. For a number field \(F\), let \(h_2(F)\), \(\text{Cl}_2(F)\) and \(F^1\) denote its 2-class number, 2-class group and Hilbert 2-class field respectively. Also, the pair \((2,2m)\) denotes the group \(\mathbb{Z}_2\times\mathbb{Z}_{2^m}\).
The main theorem is: Let \(k\) be an imaginary quadratic field with \(\text{Cl}_2(k)\cong (2,2^m)\). Then rank \(\text{Cl}_2(k^1)= 2\) if and only if \(k= \mathbb{Q}(\sqrt{-p_1p_2p_3})\), where \(p_1\), \(p_2\) and \(p_3\) are primes satisfying \(p_1\not\equiv 3\not\equiv p_2\pmod 4\) and \(p_3\equiv 3\pmod 4\), \(({p_1\over p_2})= -1\), \(({p_1\over p_3})= ({p_2\over p_3})= 1\) and \(h_2(K)= 2\), where \(K\) is a non-normal quartic subfield of one of the two unramified cyclic quartic extensions of \(k\) such that \(\mathbb{Q}(\sqrt{p_1p_2})\subset K\). For example, the fields \(\mathbb{Q}(\sqrt{-d})\) for \(d= 310, 406, 598, 1443\) and \(1615\) all satisfy these conditions.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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