On the Fueter model and monogeneity of rings of integers
Let \(L/M\) be an extension of number fields. It is said that \(L\) is monogenic over \(M\) if \({\mathfrak D}_ L={\mathfrak D}_ M[\alpha]\) for some \(\alpha\in {\mathfrak D}_ L\), where \({\mathfrak D}_ L\) denotes the ring of integers in \(L\). Let \(K\) be an imaginary quadratic field in which 2 splits. For an integral ideal \({\mathfrak f}\) let \(K({\mathfrak f})\) denote the ray class field mod \({\mathfrak f}\). \textit{Ph. Cassou-Noguès} and \textit{M. J. Taylor} [J. Lond. Math. Soc., II. Ser. 37, 63--72 (1988; Zbl 0639.12001)] proved that \(K({\mathfrak f})\) is monogenic over \(K(1)\), the Hilbert class field, and gave a generator in terms of torsion values obtained from the Fueter model if \({\mathfrak f}\) is coprime to 2. In this paper the authors remove the condition that \({\mathfrak f}\) be coprime to 2. The proof uses the minimal Fueter model instead of the usual Fueter model. The minimal Fueter model has better properties about reduction modulo primes than the usual Fueter model, while the \(x\)-coordinates of both models coincide.
scientific article; zbMATH DE number 166056