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