On Galois structure of the integers in elementary abelian extensions of local number fields
Let \(p\) be an odd prime number and \(G\) an elementary abelian group of order \(p^2\). Let \(k\) be a finite extension of \(\mathbb{Q}_p\) containing a primitive \(p\)th root of unity. Let \(O_k\) be the ring of integers of \(k\). Let \(K/k\) be a totally ramified Galois (Kummer) extension with Galois group \(G\), and \(O_K\) its ring of integers. Let \(K'/k\) be another such extension. The author gives a necessary and sufficient condition for \(O_K\) and \(O_{K'}\) to be \(O_k[G]\)-isomorphic. From this result he deduces that there exist \(K/k\) and \(K'/k\) such that \(O_K\) and \(O_{K'}\) are \(\mathbb{Z}_p[G]\)-isomorphic, but not \(O_k[G]\)-isomorphic. In the last section, the author obtains sufficient condition, in term of ramification, for \(O_K\) to be not free over its associated order in \(k[G]\).
scientific article; zbMATH DE number 5190752