# Reference request: principalization theorem

Let $K$ be a number field, and $\mathbb{I}_K$ the group of ideles. The Hilbert class field $M$ of $K$ is the class field of the open subgroup $H = K^{\ast} \mathbb{I}_K^{S_{\infty}}$, where $$\mathbb{I}_K^{S_{\infty}} = \prod\limits_{v \mid \infty} K_v^{\ast} \prod\limits_{v \nmid \infty} \mathcal O_v^{\ast}$$ From local class field theory, we have the result that $v$ is unramified in $M$ if and only if $\mathcal O_v^{\ast}$ (or $K_v^{\ast}$ if $v$ is infinite) is contained in $H$. Since $H$ is the smallest open subgroup of $\mathbb{I}_K$ which contains $K^{\ast}$ and all these local groups, we get that $M$ is the largest abelian extension of $K$ in which every place of $K$ is unramified.

It is a major theorem that every prime ideal of $K$ is principal in the Hilbert class field. Does anyone know where I can find an elementary-ish proof of this result?