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Define \( N: \mathbb{Z}[i] \rightarrow \mathbb{Z} \) by \( N(a+b i)-a^{2}+b^{2} \). (c) (4 points) Verify that \( N(x y)-N(x) N(y) \) for all \( x, y \in \mathbb{Z}[i] \). (d) (6 points) Prove or disprove that \( N \) a ring homomorphism.