Fitting ideals of class groups of real fields with prime power conductor
We quote the authors' words: ``In this paper we prove a structure result on the ideal class group and on the narrow ideal class group of totally real fields with prime power conductor. For any number field \(K\) we denote its ideal class group by \(\text{Cl}_K\), its narrow ideal class group by \(\text{Cl}_K^\infty\), its unit group by \(O_{\mathcal K}^*\) and the group of totally positive units by \(O_{{\mathcal K},+}^*\). Let \(l\) be a prime number, let \(n\in \mathbb{N}\) and suppose that \(K\) has conductor \(l^n\), that is to say \(K\) is contained in the field \(\mathbb{Q} (\zeta_{l^n})\) obtained by adjoining to \(\mathbb{Q}\) a primitive \(l^n\)th root of unity \(\zeta_{l^n}\) and \(n\) is minimal. We denote by \(G\) the Galois group \(\text{Gal} (K/\mathbb{Q})\). We define the group of cyclotomic units \(\text{Cyc}_K\) of \(K\) as \[ \text{Cyc}_K= O_{\mathcal K}^*\cap (1-\zeta_{l^n})^{\mathbb{Z}[ \text{Gal} (\mathbb{Q}(\zeta_{l^n})/ \mathbb{Q})]}. \] It is known that \(\text{Cyc}_L\) has finite index in \(O_{\mathcal K}^*\) and that the index \((O_{\mathcal K}^*: \text{Cyc}_K)\) is equal to the order of \(\text{Cl}_K\), but the two Galois modules \(\text{Cl}_K\) and \(O_{\mathcal K}^*/ \text{Cyc}_K\) are not isomorphic in general, not even as abelian groups. In this paper we prove that their Fitting ideals \(\text{Fit}_{\mathbb{Z}[G]} (\text{Cl}_K)\) and \(\text{Fit}_{\mathbb{Z}[G]} (O_{\mathcal K}^*/ \text{Cyc}_K)\) are equal. We also prove a result about the \(\mathbb{Z}[G]\)-Fitting ideal of the narrow ideal class group \(\text{Cl}_K^\infty\).
scientific article; zbMATH DE number 1247931