Anneaux d'entiers dans le même genre
Let \({\bar {\mathbb{Q}}}\) denote an algebraic closure of \({\mathbb{Q}}\), the field of rational numbers. If N is a finite extension of \({\mathbb{Q}}\), then \(G_ N\) denotes the Galois group Gal(\({\bar {\mathbb{Q}}}/N)\) and \({\mathbb{Z}}_ N\) denotes the ring of integers of N. For a finite group \(\Gamma\), \(\epsilon\) (\(\Gamma)\) is the set of extensions N of \({\mathbb{Q}}\) such that \(\Gamma\) is isomorphic to a subgroup of the automorphism group of N over \({\mathbb{Q}}\). If \(N\in \epsilon (\Gamma)\) then \({\mathbb{Z}}_ N\) has a \({\mathbb{Z}}[\Gamma]\)-structure of rank \(r=[N^{\Gamma}:{\mathbb{Q}}].\) The following result (conjectured by Wilson) is proved: If N,N'\(\in \epsilon (\Gamma)\), such that \({\mathbb{Z}}_ N\) is locally isomorphic to \({\mathbb{Z}}_{N'}\), as \({\mathbb{Z}}[\Gamma]\)-modules, then there exists a \({\mathbb{Z}}[\Gamma]\)-module X of finite type and without \({\mathbb{Z}}\)-torsion and an isomorphism of \({\mathbb{Z}}[\Gamma]\)-modules between \({\mathbb{Z}}_ N\oplus X\cong {\mathbb{Z}}_{N'}\oplus X\).
scientific article; zbMATH DE number 3879039