Recorded, not proved: every finite abelian extension of $\mathbb{Q}$ lies in a cyclotomic field

Kronecker–Weber theorem. Let $L\mathrm{/}\mathbb{Q}$ be a finite Galois extension whose Galois group is abelian. Then there is an integer $n\ge 1$ with

$L\subseteq \mathbb{Q}({\mu }_n)$

(The cyclotomic extension $K({\mu }_n)$ as a splitting field of $t^n-1$).

The statement fails over larger base fields. Over $K=\mathbb{Q}(i)$ the extension obtained by adjoining a fourth root of $1+i$ is abelian over $K$ and is contained in no cyclotomic extension of $K$.

What this library does prove. The converse half is Every intermediate field of $\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q}$ is Galois over $\mathbb{Q}$ with abelian Galois group: every intermediate field of $\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q}$ is Galois over $\mathbb{Q}$ with abelian Galois group. Together with $[\mathbb{Q}({\zeta }_n):\mathbb{Q}]=\phi (n)$ and $\mathrm{Gal}(\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q})\cong (\mathbb{Z}\mathrm{/}n{)}^{\times }$, which computes $\mathrm{Gal}(\mathbb{Q}({\mu }_n)\mathrm{/}\mathbb{Q})$ exactly, that half says the subfields of cyclotomic fields are abelian; Kronecker–Weber says there are no others.

What would prove it, and which track that belongs to. The standard argument reduces the global statement to a local one at each prime that ramifies in $L$, and then analyses the higher ramification groups of the $p$-adic rationals to show that the local extension is contained in a local cyclotomic extension. Every ingredient of that reduction — valuations, local fields, the ring of integers of a number field, decomposition and inertia groups — belongs to algebraic number theory, and none of it is developed anywhere in this library. An alternative route through class field theory needs strictly more. Neither source consulted here proves the theorem: Conrad calls it deep and states it without proof, and Milne states it only in a footnote to an exercise.

Why it is recorded rather than omitted. The arithmetic consequences of the theorem are stated elsewhere in terms of it, so a page that builds the cyclotomic machinery and then says nothing about its sharpest classical application would leave that seam pointing at nothing. Recording it with the fuchsia not-proved-here marking is the honest form: the reader is told, at the point of contact, that the proof belongs to a later algebraic-number-theory track.