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 not-equal q and both odd, then K* = F1* ... F(n)*. These results are derived from calculations involving modules over the integral group ring Z[G]. (C) 1994 Academic Press, Inc.