Cubic fields with index 2
Let \(K=\mathbb Q(\alpha)\) be a cubic field, where \(\alpha\) is an integer of \(K\). The index of \(\alpha\) is \(i(\alpha)=(d(\alpha)/d(K))^{\frac 12}\), where \(d(K)\) is the discriminant of \(K\). The minimal index of \(K\) is \(m(K)=\min\{i(\alpha)\}\), and the index of \(K\) is \(i(K)=\text{gcd} \{i(\alpha)\}\), where \(\alpha\) ranges through all integral generators of \(K\). Let \(M\) denote the splitting field of \(K\). In an earlier paper [Rocky Mt. J. Math. 31, 1103-1109 (2001; Zbl 0992.11059)] the authors proved that there exist infinitely many \(K\) with \(m(K)=1\) such that \(M\) contains a given quadratic field. In the present paper they prove: Let \(d\in\mathbb Z\) be squarefree. (a) If \(d\not\equiv 1\mod 8\), there is no \(K\) with \(i(K)=2\) and \(\mathbb Q(\sqrt d)\subset M\). (b) If \(d\equiv 1\mod 8\), then there exist infinitely many \(K\) with \(i(K)=m(K)=2\) and \(\mathbb Q(\sqrt d)\subset M.\)
scientific article; zbMATH DE number 1747323