Steinitz classes of relative Galois extensions of 2-power degree and embedding problems
Let \(K/k\) be a normal extension of number fields of degree \(n\), and let \({\mathcal O}_K\) denote the ring of integers of \(N\). As an \({\mathcal O}_k\)-module, it is isomorphic to \({\mathcal O}_k^{n-1} \oplus I\) for some ideal \(I\) in \({\mathcal O}_k\), whose ideal class is called the Steinitz class of the extension \(K/k\). The author continues recent investigations of the question of which ideal classes of \(k\) can be realized as Steinitz classes as \(N\) runs through certain classes of tamely ramified extensions of \(k\) with given Galois group. His main results are the following: every ideal class \(c\) of \(k\) can be realized as the Steinitz class of one of the following extensions: 1) a quadratic extension of \(k\) that can be embedded into a tamely ramified cyclic quartic extension of \(k\); 2) a biquadratic extension of \(k\) that can be embedded into a tamely ramified quaternion extension of degree \(8\) over \(k\); 3) a tamely ramified cyclic quartic extension; 4) a tamely ramified quaternion extension of \(k\) of degree \(8\). For 2) he has to assume that \(k\) contains a fourth root of unity \(i\) or that \(k(i)/k\) is ramified, while for 3) and 4) he assumes that the class number of \(k\) is odd. These results are proved by realizing these extensions more or less explicitly as ray class fields.
scientific article; zbMATH DE number 1274758