Conductor ideals of orders in algebraic number fields
Let \(K\) be a number field, \(\mathcal O_K\) its ring of algebraic integers and let \(\mathcal O\) be an order of \(K\). Let us recall that the conductor \(\mathfrak f\) of \(\mathcal O\) is defined by \(\mathfrak f=\{x\in K:\;x\mathcal O_K\subseteq \mathcal O\}\). In the paper under review the authors give a new version of the proofs of results from \textit{Ph. Furtwängler} [Anzeiger Wien 56, 75 (1919; JFM 47.0146.04)] about necessary and sufficient conditions for an ideal of \(\mathcal O_K\) to be the conductor of some order of \(K\). More concretely, first of all it is shown that it is enough to study the case in which the norm of \(\mathfrak f\) is the power of a rational prime. Under this assumption, \(\mathfrak f=\mathfrak p_1^{k_1}\cdots\mathfrak p_g^{k_g}\) is shown to be a conductor if and only if for every \(1\leq i\leq q\) we have: if \(\mathfrak p_i\) is of absolute degree \(1\) and \(k_i\equiv 1\pmod{e_i}\), then there exists \(j\neq i\) such that \(k_j>\frac{k_i-1}{e_i}e_j\), where \(e_i\) is the ramification index of \(\mathfrak p_i\) over \(\mathbb Q\).
scientific article; zbMATH DE number 6353785