ON SOME PROPERTIES OF GALOIS GROUPS OF UNRAMIFIED EXTENSIONS

Let k be an algebraic number field of finite degree and koo be the maximal cyclotomic extension of k. Let (L) over tilde (k) and L-k be the maximal unramified Galois extension and the maximal unramified abelian extension of k(infinity) respectively. We shall give some remarks on the Galois groups Gal((L) over tilde (k)/k(infinity)), Gal(L-k/k(infinity)) and Gal((L) over tilde (k)/k). One of the remarks is concerned with non-solvable quotients of Gal((L) over tilde (k)/k(infinity)) when k is the rationals, which strengthens our previous result.