Existence of relative integral basis over quadratic fields and Pólya property
For an algebraic number field \(K\), \textit{H. B. Mann} [Proc. Amer. Math. Soc. 9, 167--172 (1958; Zbl 0081.26602)] gave a criterion for a quadratic extension \(L\) of \(K\) to have an integral basis over \(K\). Specifically, \(L\) has an integral basis over \(K\) if and only if \(L=K(\sqrt{\alpha})\) for some \(\alpha\) for which \((\alpha)\) is the discriminant of \(L\) over \(K\). From this, he showed that every quadratic extension \(L\) of \(K\) has an integral basis over \(K\) if and only if \(K\) has class number 1 (see also \textit{W. Narkiewicz} [Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag (1990; Zbl 0717.11045)], Corollary to Proposition 7.16, p. 403).\par \textit{R. H. Bird} and \textit{C. J. Parry} [Pac. J. Math. 66, 29--36 (1976; Zbl 0356.12003)] later considered the special case \(K=\mathbb{Q}(\sqrt{m})\) and \(L=\mathbb{Q}(\sqrt{m},\sqrt{n})\) in detail, where \(m,n\) are distinct squarefree rational integers. In this case, explicit and concrete conditions under which \(L\) has an integral basis over \(K\) were given, and an explicit integral basis was produced whenever one exists. In the cases where \(K\) is imaginary or \(K\) is real and has a unit of norm \(-1\), the conditions involve only rational congruences on \(m\) and \(n\). In the remaining case that \(K\) is real and has only units of norm \(+1\), the conditions sometimes also involve the fundamental unit.\par In the paper under review, the following theorem is proven. Let \(K=\mathbb{Q}(\sqrt{m})\) be a quadratic field. Then every biquadratic field \(L=\mathbb{Q}(\sqrt{m},\sqrt{n})\), for every squarefree integer \(n\) distinct from \(m\), has a relative integral basis over \(K\) if and only if \(K\) is a Pólya field, or equivalently that all strongly ambiguous ideal classes of \(K\) are trivial.\par For a general number field \(F\), the Pólya property can be characterized in various ways. For example, let \(\mathcal{O}_F\) denote the ring of integers of \(F\) and let \[ \text{Int}(\mathcal{O}_F) = \{f\in F[X]: f(\mathcal{O}_F)\subseteq \mathcal{O}_F\} \] be the the ring of integer-valued polynomials on \(\mathcal{O}_F\). Then \(\text{Int}(\mathcal{O}_F)\) is a free \(\mathcal{O}_F\)-module, and \(F\) is said to be Pólya if and only if \(\text{Int}(\mathcal{O}_F)\) has a basis consisting of exactly one polynomial of each degree. This property can be shown to be equivalent to the principality of all so-called Ostrowski ideals of \(F\).
scientific article; zbMATH DE number 7377610