Multiplicative groups of Galois extensions (Q1327023)

The author obtains the following interesting results: Theorem. Let \(K\) be a Galois extension of \(k\) with Galois group \(G\) and suppose that \(G\) contains two dihedral subgroups \(D_ p\) and \(D_ q\) for distinct odd primes \(p\), \(q\). Then if \(F_ 1,\dots, F_ N\) are the maximal proper subfields of \(K/k\) then \(K^ \times= F_ 1^ \times \dots F_ N^ \times\). Corollary. If \(K\) is a Galois extension of \(k\) with Galois group \(S_ n\), \(n\geq 5\), then \(K^ \times\) is generated by the multiplicative subgroups of proper intermediate fields.