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.

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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.

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(Solved): Define \( N: \mathbb{Z}[i] \rightarrow \mathbb{Z} \) by \( N(a+b i)-a^{2}+b^{2} \). (c) (4 points) ...

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