it's actually an if and only if statement. the reason is quite clear: the ring of integers of , is a UFD if and only if it's a PID because is always a Dedekind domain.

on the other hand by definition the ideal class group of is trivial if and only if every ideal of is principal, i.e. if and only if is a PID. so the class

number of is 1 if and only if is a PID. finally the dimension of the Hilbert class field of as a vector space over is exactly and the result follows.