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Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur as Galois group but I don't know how the "unramified" condition plays a role.

Also I'd like to know if there is a way of constructing those extensions.

I appreciate every help, thanks, Tom :-)

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The maximal unramified abelian extension of $K$ is called the Hilbert class field of $K$; it is finite over $K$, and its Galois group over $K$ is isomorphic with the ideal class group of $K$.

The maximal abelian unramified $p$-extension is therefore the subfield of $H$ fixed by the complement of the $p$-part of the ideal class group of $K$. Its Galois group over $K$ is isomorphic with $\text{Cl}(K) \otimes_\mathbf Z \mathbf Z_p$ (the $p$-part of $\text{Cl}(K)$).

Constructing these extensions is a complicated thing. It is known as "explicit class field theory". If you'd like to do it for one number field in particular, perhaps SAGE can help you. If you'd like to learn how to do it in general, you'll have to pick up a book on class field theory.

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  • $\begingroup$ Thank you for your useful comment! I'm totally aware of the theorems from class field theory, but it was nevertheless enlightening to hear it again. Actually I know that the $p$-part of the ideal class group, which I denoted by $_p Cl(K)$ is not trivial and that gives me this unramified abelian finite field extension, of which I'd like to study the Galois group. But except for knowing that it is a finite abelian $p$-group, I don't know how the "unramified"-part translates into group theory. Is there a translation? $\endgroup$ – BIS HD Oct 25 '13 at 8:46
  • $\begingroup$ Dear @BISHD I'm not exactly sure what you mean by "I don't know how the "unramified"-part translates into group theory. Is there a translation?"... $\endgroup$ – Bruno Joyal Oct 25 '13 at 14:05
  • $\begingroup$ That is exactly what I'm asking :) But maybe there is no translation - So in fact my feeling that every finite abelian unramified $p$-extension has a Galois group isomorphic to $\mathbb{Z} / p^n \mathbb{Z}$ is wrong I guess?! $\endgroup$ – BIS HD Oct 25 '13 at 14:11
  • $\begingroup$ @BISHD Yes, it's wrong. The $p$-part can be pretty complicated. $\endgroup$ – Bruno Joyal Oct 26 '13 at 0:32
  • $\begingroup$ To see how "pretty complicated" it can be, consider the cyclotomic field K_0 = Q (mu_p), p an odd prime, as well as the tower of extensions K_n = Q (mu_p^n), and write A_n for the p-class group of $K_n$. The minus parts (under complex conjugation) of the A_n 's, after passing to the inverse limit w.r.t norm maps, is related by Iwasawa theory to the p-adic zeta function which interpolates the special values of the complex zeta function of K+, the maximal real subfield of K . As for the + part of K_0, Vandiver's conjecture (unsolved) predicts its triviality . $\endgroup$ – nguyen quang do Apr 18 '16 at 6:08

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