On \(S\)-units almost generated by \(S\)-units of subfields (Q2542318)

Let \(K/k\) be a finite Galois extension of number fields, \(S\) a finite set of primes of \(K\), and \(\Phi\) a set of intermediate fields (\(S\) and \(\Phi\) are closed under the action of the Galois group). Necessary and sufficient conditions are given for the \(S\)-units in fields of \(\Phi\) to generate a subgroup of finite index in the \(S\)-units of \(K\). the proof is based on the Frobenius reciprocity law. These are then applied when there is a totally split prime in \(S\), to some examples of \(\Phi\).