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 ].
