Determinants of integral ideal lattices and automorphisms of given characteristic polynomial
Let \(K\) be an algebraic number field with a nontrivial involution \(\alpha\to \overline\alpha\). An integral ideal lattice in \(K\) is a fractional ideal \({\mathfrak A}\) of \(K\) together with a quadratic form \(b(x,y)=Tr (\alpha\times \overline y)\) with values in \(\mathbb{Z}\) for \(x,y\in {\mathfrak A}\), where \(\alpha\) is an element of \(K^\times\) with \(\alpha= \overline\alpha\). The author gives a characterisation of the determinants and signatures of integral ideal lattices over a given algebraic number field. This is then used to obtain an existence criterion for lattice automorphisms of given characteristic polynomials. In particular the author gives a different proof of a result of \textit{B. Gross} and \textit{C. McMullen} [J. Algebra 257, 265-290 (2002; Zbl 1022.11016)].
scientific article; zbMATH DE number 1868061