Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr] we write K([lscr]) for the subfield of K(a) obtained by adjoining to K all [lscr]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr][ratio ]G(K)→Z*[lscr] for the cyclotomic character defining the Galois action on all [lscr]-power roots of unity. We write χ[lscr]=χ[lscr] mod [lscr][ratio ]G(K)→Z*[lscr]→(Z/[lscr]Z)* for the cyclotomic character defining the Galois action on the [lscr]th roots of unity. The character χ[lscr] identifies Gal (K([lscr])/K) with a subgroup of Z*[lscr]=Gal (Q([lscr])/Q). Let μ(Z[lscr]) be the finite cyclic group μ(Z[lscr]) of all roots of unity in Z*[lscr]. Its order is equal to [lscr]−1 if [lscr] is odd and 2 if [lscr]=2. Let Q([lscr])′ be the subfield of μ(Z[lscr])-invariants in Q([lscr]). Clearly, Gal (Q([lscr])/Q([lscr])′)=μ(Z[lscr]) and Gal (Q([lscr])′/Q)=Z*[lscr]/μ(Z[lscr]) is isomorphic to Z[lscr].