Hilbert-Speiser number fields and Stickelberger ideals (Q988066)

Let \(p\) be a prime number. For a number field \(F\), let \(\mathcal{O}_{F}\) be the ring of integers and let \(\mathcal{O}_{F}'=\mathcal{O}_{F}[1/p]\) be the ring of \(p\)-integers of \(F\). A finite Galois extension \(N/F\) with group \(\Gamma\) is said to have a normal \(p\)-integral basis (\(p\)-NIB for short) if \(\mathcal{O}_{N}'\) is free (necessarily of rank \(1\)) as a module over the group ring \(\mathcal{O}_{F}'[G]\). We say that \(F\) satisfies the condition (\(H_{p^{n}}'\)) when every abelian extension \(N/F\) of exponent dividing \(p^{n}\) has a \(p\)-NIB, and that \(F\) satisfies (\(H_{p^{\infty}}'\)) when it satisfies (\(H_{p^{n}}'\)) for all \(n\). The classical Hilbert-Speiser Theorem implies that the rationals \(\mathbb{Q}\) satisfy (\(H_{p^{\infty}}'\)) for all \(p\). The article under review is motivated by the question of whether any number field \(F \neq \mathbb{Q}\) satisfies (\(H_{p^{\infty}}'\)) for some \(p\). The main result of the article under review is a necessary and sufficient condition for a number field \(F\) to satisfy (\(H_{p^{n}}'\)) for a given \(n \in \mathbb{N}\). For \(1 \leq i \leq n\), let \(K_{i}=F(\zeta_{p^{i}})\) where \(\zeta_{p^{i}}\) is a primitive \(p^{i}\)-th root of unity. Let \(Cl_{K_{i}}'\) denote the ideal class group of \(\mathcal{O}_{K_{i}}'\). Let \(\mathcal{S}_{F,i}\) denote a certain ``Stickelberger ideal'', which we shall not define here. Then \(F\) satisfies (\(H_{p^{n}}'\)) if and only if \(\mathcal{S}_{F,i}\) annihilates \(Cl_{K_{i}}'\) for all \(1 \leq i \leq n\). As an application, a candidate is given for an imaginary quadratic field which has the possibility of satisfying (\(H_{p^{\infty}}'\)) for a small prime \(p\).