The Kronecker—Weber Theorem

The Kronecker—Weber theorem asserts that every abelian extension of the rationals is contained in a cyclotomic field. It was first stated by Kronecker in 1853, but his proof was incomplete. In particular, there were difficulties with extensions of degree a power of 2. Even in the proof we give below this case requires special consideration. The first proof was given by Weber in 1886 (there was still a gap; see Neumann [1]). Both Kronecker and Weber used the theory of Lagrange resolvents. In 1896, Hilbert gave another proof which relied more on an analysis of ramification groups. Now, the theorem is usually given as an easy consequence of class field theory. We do this in the Appendix. The main point is that in an abelian extension the splitting of primes is determined by congruence conditions, and we already know that p splits in $$ \mathbb{Q}\left( {{\zeta _n}} \right)$$ if $$ p \equiv 1$$ and only if mod n.