Multiplicative Groups of Galois Extensions Original Research Article

Suppose that K is Galois over k with group G, and suppose that F1 … Fn are maximal among the intermediate subfields. Then it is shown that if G=Dp, 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 Dp and Dq, p ≠ q and both odd, then K* = F*1 … F*n. These results are derived from calculations involving modules over the integral group ring image[G].