TITLE: Finite abelian unramified $p$-extension of a number field
QUESTION [6 upvotes]: 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 :-)

REPLY [8 votes]: 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.