On calculating the number \(N(D)\) of global cubic fields \(F\) of given discriminant \(D\)
Let \(F\) be a global cubic field, i.e. a field extension \(F/F_0\) of degree \(3\), where \(F_0=\mathbb{Q}\) or \(F_0=\mathbb{F}_q(t)\). For given integral \(D\), i.e. \(D \in \mathbb{Z}\) or \(D \in \mathbb{F}_q [t]\), resp., let \(N(D)\) denote the number of non-isomorphic global fields \(F\) with discriminant \(D\). The method to determine \(N(D)\) uses class field theory and goes back to \textit{H. Hasse} [Math. Z. 31, 565--582 (1930); correction ibid. 31, 799 (1930; JFM 56.0167.02)] in the case of number fields and to the second author [Publ. Math. Debr. 79, No. 3--4, 611--621 (2011; Zbl 1249.11112)] in the case of function fields. The authors present a joint treatment for number fields and function fields, and first recall the results for Galois extensions \(F/F_0\). The calculation of \(N(D)\) in the non-Galois case is given in Theorem 3.3 and requires heavy computational methods. In Section 4 the authors present some new ideas and a new algorithm, which needs only calculations in the quadratic subfield of the Galois closure of \(F\). This makes it possible to obtain \(N(D)\) for much larger values of \(D\) than hitherto.
scientific article; zbMATH DE number 7493035