It is and it isn't. For example, you can adjoin sqrt(-1). This time there are finitely many units, but still unique factorisation. Instead adjoin sqrt(-17) and

It is and it isn't. For example, you can adjoin sqrt(-1). This time there are finitely many units, but still unique factorisation. Instead adjoin sqrt(-17) and you get finitely many units but you also do not get unique factorisation.