Construction of 3-Hilbert class field of certain imaginary quadratic fields (Q963131)

In this short paper, the author describes a way of finding the 3-class field \(H_k\) of an imaginary quadratic field \(k=\mathbb Q(\sqrt{-d})\). As usual, one assumes \(d\) squarefree. The method is simply Kummer theory over \(k_z=k(\zeta_3)\) plus descent from \(k_z\) to \(k\). The main result (theorem 3) exhibits \(H_k\) as \(k(\root 3 \of {\alpha^2}+\root 3 \of {\alpha^{-2}})\) with \(\alpha\) the fundamental unit of \(\mathbb Q(\sqrt{3d})\), under the following three assumptions: (1) \(d\) is not congruent to 3 modulo 9, (2) \(k\) has class number 3, and (3) \(\mathbb Q(\sqrt{3d})\) has class number 1. As an example, the author treats the case \(d=23\), which is actually the smallest value possible. Reviewer's remarks: (a) The hypothesis (1) is superfluous: if \(d\) is 3 (or 6, for that matter) modulo 9, then \(d\) is divisible by 3; by genus theory, since \(h_k\) is odd by hypothesis (2), \(d\) can have no other prime factor, so \(d=3\); but this is impossible, again by (2). (b) Hypotheses (2) and (3) seem rather restrictive. It would suffice to assume \(h_k\) divisible by 3 and not by 9, and \(h_{{\mathbb Q}(\sqrt{3d})}\) prime to 3; one would in fact expect this since \(H_k\) is defined to be the 3-class field, not the full Hilbert class field. As it stands, the main theorem would only cover finitely many imaginary quadratic fields. (c) It seems likely that a slightly more direct argument is possible. Under the author's hypotheses, the extension \(H_kk_z/k_z\) must be given by adjoining a cube root of a unit \(\beta\) of \(k_z\), since \(k_z\) will have class number prime to 3. Easy arguments show that we may assume \(\beta= \zeta\alpha\) with \(\alpha\) as in the first paragraph and \(\zeta\) a third root of unity. The usual descent argument will give exactly the desired statement, once we know that \(\zeta=1\). This can be done by a calculation. The point to check is: if \(\alpha\) itself is not hyperprimary (i.e., congruent to a cube modulo the third power of the prime above 3), then \(\zeta_3\alpha\) and \(\zeta_3^2\alpha\) will not be hyperprimary either; and adjunction of a cube root gives an extension unramified at 3 iff the radical is hyperprimary. (d) In particular, one can replace \(\alpha^2\) by \(\alpha\) in the example \(d=23\), leading to a ``smaller'' generating polynomial \(x^3-3x-25\). Actually, by more general arguments, any monic cubic with integer coefficients and discriminant \(-23\) times a square will work; the ``smallest'' such is \(x^3-x-1\), as is fairly well known; this has discriminant \(-23\), which is in fact the minimum in absolute value for \textit{all } such cubics.