Multiplicative groups of fields modulo products of subfields
Let \(K/k\) be a finite separable field extension. Let \(E_1, \dots, E_r\) be intermediate fields of \(K/k\). In this paper some properties of the quotient \(K^*/ E^*_1 \dots E^*_r\) are investigated. If \(r\leq 2\), then the \(n\)-torsion subgroup of \(K^*/ E^*_1 \dots E^*_r\) is finite for any \(n>0\). If \(E_1\) is finitely generated over its prime field, then \(K^*/ E^*_1\) is finite. Let \(\mu (K)\) denote the group of roots of unity contained in \(K\). Then \(K^*/ E^*_1 E^*_2\) is finite if and only if \(\mu (K)/ \mu(E_1) \mu(E_2)\) is finite. These results are no longer true if \(r>2\). Examples are given that \(K^*/ E^*_1 \dots E^*_r\) is trivial, finite, infinite torsion or has infinite torsion free rank. Some of these statements or examples are already contained in previous papers of the authors. The aim of the paper under review is a general approach to these finiteness results. The authors start with the general concept of \(G\)-modules and then they apply Galois cohomology and things like Hilbert 90.
scientific article; zbMATH DE number 858645
scientific article