# Gaussian integers, ring homomorphism and kernel

## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$....

I dont know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$

## Answers and Replies

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## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$....

Maybe bd=-bd (mod 2) ??

I dont know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$
Yes, so take $a+bi \in \ker(\varphi$. Then $\varphi(a+bi)=0$. Now just write things out using the definition of $\varphi$.