The Kronecker-Weber theorem, sometimes known as the Kronecker-Weber-Hilbert theorem, is one of the earliest known results in class field theory.
In layman's terms, the Kronecker-Weber theorem says that cyclotomic extensions of the field 
of rational numbers capture
in a very precise way all the finite abelian
extensions of 
. More precisely, the theorem says that every finite abelian extension of 
is contained in a cyclotomic
extension, i.e., that for an arbitrary finite abelian
Galois extension 
, there exists an integer 
for which 
where 
is a primitive root of unity.
From a more computational viewpoint, the theorem can be rephrased to say that every algebraic integer whose Galois
group is abelian can be expressed as a sum
of 
th
roots of unity for some 
.
Incomplete proofs of the theorem were given first by Kronecker and by Weber in 1853 and 1886, respectively. The first complete proof
was given by Hilbert in 1896, who later hinged the twelfth of what have come to be
known as Hilbert's problems on the desire to
find generalizations of the theorem for fields other than
,
whereby sufficient analogues of roots of unity and
holomorphic functions in several variables
would also be necessary (Holzapfel 1995).
