Multiplicative groups of galois extensions

Suppose that K is Galois over k with group G, and suppose that F₁ … F_n are maximal among the intermediate subfields. Then it is shown that if G=D_p, p an odd prime, then K*/F1* … F*_n is a subgroup of F*/k* · (F*)^p where F is the unique proper Galois subfield. One then deduces that if G contains two dihedral groups D_p and D_q, p ≠ q and both odd, then K* = F*₁ … F*_n. These results are derived from calculations involving modules over the integral group ring Z[G].