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 abelian extensions
of 
.
More precisely, the theorem says that every abelian
extension of 
is contained in a cyclotomic
extension, i.e., that for an arbitrary 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).
