Proof. This will first be proven for
cyclic.
Let
. By Dirichlet's theorem on primes in arithmetic progressions, there exists a prime
with
. Let
denote a primitive
root of unity. Let
. Then
is Galois with
cyclic of order
. Since
divides
, there exists a subgroup
of
such that
. Since
is cyclic, it is abelian, and
is a normal subgroup of
. Let
, the subfield of
fixed by
. Then
is Galois with
cyclic of order
. Thus,
.
Let
and
be distinct primes with
and
. Then there exist subfields
and
of
and
, respectively, such that
and
. Note that
since
. Thus,
. Therefore, for every prime
with
, there exists a distinct number field
such that
is Galois and
. The theorem in the cyclic case follows from using the full force of Dirichlet's theorem on primes in arithmetic progressions: There exist infinitely many primes
with
.
The general case follows immediately from the above argument, the fundamental theorem of finite abelian groups, and a theorem regarding the Galois group of the compositum of two Galois extensions. 