If a and b are ideals in the ring of integers O of a quadratic number field, show that $\displaystyle N(a*b) = N(a) * N(b)$

Recall Norm: $\displaystyle N(a):= [O:a]$ (order of gp. O/a).

Moreover if $\displaystyle a \in O$, and a' is the conjugate ideal of a, show that $\displaystyle a * a' = (N(a))$, the principal ideal generated by $\displaystyle N(a)$

Recall Conjugate ideal: $\displaystyle a' = \{x' : x \in a \}$ (x' is conjugate of x in O).