Suppose $\displaystyle p $ is irreducible in $\displaystyle \mathbb{Z}[i] $.

1. Let $\displaystyle A_n = (p^n) $ (the ideal generated by $\displaystyle p^n $). Prove $\displaystyle \mathbb{Z}[i]/(p) \cong A_n/A_{n+1} $ as additive abelian groups.

2. Show $\displaystyle |\mathbb{Z}[i]/(p^n)| = |\mathbb{Z}[i]/(p)|^n $.