# Math Help - Hilbert class field

1. ## Hilbert class field

For the number fields K whose ring of integer is UFD,
why the Hilbert class field of K is K itself???

thanks

2. Originally Posted by chipai
For the number fields K whose ring of integer is UFD,
why the Hilbert class field of K is K itself???

thanks
it's actually an if and only if statement. the reason is quite clear: $\mathcal{O}_K,$ the ring of integers of $K$, is a UFD if and only if it's a PID because $\mathcal{O}_K$ is always a Dedekind domain.

on the other hand by definition $H(K),$ the ideal class group of $K,$ is trivial if and only if every ideal of $\mathcal{O}_K$ is principal, i.e. if and only if $\mathcal{O}_K$ is a PID. so $|H(K)|,$ the class

number of $K,$ is 1 if and only if $\mathcal{O}_K$ is a PID. finally the dimension of the Hilbert class field of $K,$ as a vector space over $K,$ is exactly $|H(K)|$ and the result follows.