Steinitz classes of extensions with Galois group \(A_4\)
Let \(k\) be a number field and \(E/k\) a cyclic extension of degree \(3\). The authors show that if \(k\) has odd class number, then the set \(R\) of Steinitz classes of the rings of integers \(O_N\) of extensions \(N/k\) containing \(E/k\) and with Galois group \(A_4\) is equal to the coset \(\text{ cl}(O_E)(N_{E/k}\text{ Cl}(E))^3\). Here \(\text{ cl}(O_E)\) is the Steinitz class of \(O_E\), and \(N_{E/k}\) is the relative norm applied to the ideal class group of \(E\).
scientific article; zbMATH DE number 1943337
scientific article