# Finite abelian unramified $p$-extension of a number field

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 :-)

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 $\text{Cl}(K)$).
• 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? – BIS HD Oct 25 '13 at 8:46
• 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?! – BIS HD Oct 25 '13 at 14:11
• @BISHD Yes, it's wrong. The $p$-part can be pretty complicated. – Bruno Joyal Oct 26 '13 at 0:32
• 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 . – nguyen quang do Apr 18 '16 at 6:08