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

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

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

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