I think everyone assume that it is very obvious, but actually I do not see why the following statement is obvious:
Class number 1 is equal that we have a UFD
So I tried to prove this for me:
$\rightarrow$ Class number 1 means every fractional ideal is a fractional principal ideal, so we have a PID and hence a UFD.
$\leftarrow$ We have a PID so a unique factorization into prime elements/ideals. And now I do not see how to go on.
Any suggestions? I hope it is not to obvious...