On the units of unramified cyclic cubic extensions of some subfields of \(\mathbb Q\left(\sqrt d,\sqrt{-3}\right)\)
Let \(d\) be a positive square-free integer. This article concerns the global units in cubic Galois extensions of the field \(k\), where \(k\) is either \(\mathbb Q(\sqrt d)\) or \(\mathbb Q(\sqrt d,\sqrt{-3})\). To be more precise, let us first consider the case where \(k=\mathbb Q(\sqrt d)\). We take \(K\) to be a cubic Galois extension of \(k\) which is unramified over \(k\) and dihedral over \(\mathbb Q\). According to [\textit{N. Moser}, Unités et nombre de classes d'une extension diédrale de \(\mathbb Q\), Journées Arithmétiques de Bordeaux, Astérisque 24--25, 29--35 (1975; Zbl 0313.12002)], the structure of the group of units in \(K\) can take one of five forms determined by certain unit and norm indices. The present article denotes these five structures \(\alpha,\beta,\gamma,\delta\) and \(\varepsilon\). The main result, Théorème 2.6, says that if the 3-class group of \(K\) is isomorphic to \(\mathbb Z/3\mathbb Z\times\mathbb Z/3\mathbb Z\), then in fact only two of the five structures are possible, those denoted \(\alpha\) and \(\delta\). It further states that, under these assumptions, \(\delta\) is equivalent to each of the (equivalent) statements (i) The extension \(K/k\) admits a non-trivial strongly ambiguous ideal class. (ii) \(K\) admits a Minkowski unit. (We remark that there appears to be an error in the statement of Théorème 2.6, since it claims that only the structures \(\gamma\) and \(\delta\) are possible, whereas reading the rest of the paper and the proof of that theorem, it is clear that \(\gamma\) should be replaced by \(\alpha\).) A number of computer-generated examples are given where \(K\) and \(k\) are made explicit and the isomorphism type \((\alpha\) or \(\delta)\) of the units in \(K\) determined. In particular, this allows one to read off fields \(K\) for which there exists a Minkowski unit. Attending to the case where \(k=\mathbb Q(\sqrt d,\sqrt{-3})\), the article again takes a cubic Galois extension of \(k\) which is unramified over \(k\) but this time dihedral over \(\mathbb Q(\sqrt{-3})\). Similar unit index equations to those in the case discussed above are derived, and the authors show that in this case there are four possibilities. Again, under a particular technical assumption, it is shown that only two of those possibilities can occur.
scientific article; zbMATH DE number 5554660