The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field.
In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients.
Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field.
The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity.
The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree a power of 2.