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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?

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It's in German, but a 3 page proof is given by Witt here (pages 71-73).

Ultimately Emil Artin showed that the result can be reduced to a purely group theoretic statement (the one contained in the paper linked above). See the discussion in Chapter V, Section 3 of Milne's class field theory notes.

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    $\begingroup$ I seem to remember that there's a beautiful proof in Zassenhaus's book on group theory,. $\endgroup$ – franz lemmermeyer Jul 28 '15 at 19:49
  • $\begingroup$ You have of course the chapter XIII, §4 of Artin-Tate "Class Field Theory", but can it be considered as elementary-ish ? $\endgroup$ – nguyen quang do Jan 30 '16 at 16:23

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