Let $$I_k=\lbrace ideals \ \ in \ \ \mathcal{O_k}\rbrace $$
and $$ P_k=\lbrace principal \ \ ideals \ \ in \ \ \mathcal{O_k}\rbrace \subset I_k$$
unique factorisation would follow from the equality $I_k=P_k$ .
One can also define this for fractional ideals : $$F_k=\lbrace fractional \ \ ideals \ \ in \ \ \mathcal{O_k}\rbrace $$
and$$PF_k=\lbrace principal \ \ fractional \ \ ideals \ \ in \ \ \mathcal{O_k}\rbrace $$
The class group is defined as $C_k=F_k/PF_k$
If the class group is trivial then $F_k=PF_k$ .
Why does it then follow that $I_k=P_k$ ?