# Kronecker–Weber theorem

In algebraic number theory, the Kronecker–Weber theorem states that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity. For example,

$\sqrt{5} = e^{2 \pi i / 5} - e^{4 \pi i / 5} - e^{6 \pi i / 5} + e^{8 \pi i / 5}.$

The theorem is named after Leopold Kronecker and Heinrich Martin Weber.

## Field-theoretic formulation

The Kronecker–Weber theorem can be stated more abstractly in terms of fields and field extensions. In this formulation, it states that every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.

For a given abelian extension K of Q there is a minimal cyclotomic field that contains it. 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. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory.

## History

The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree a power of 2. Weber (1886) published a proof, but this had some gaps and errors that were pointed out and corrected by Neumann (1981). The first complete proof was given by Hilbert (1896).

## Generalizations

Lubin and Tate (1965, 1966) proved the local Kronecker–Weber theorem which states that any abelian extension of a local field can be constructed using cyclotomic extensions and Lubin–Tate extensions. Hazewinkel (1975), Rosen (1981) and Lubin (1981) gave other proofs.

Hilbert's twelfth problem asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields.