Gaussian integers, ring homomorphism and kernel

[rayman123]

Homework Statement

let \varphi:\mathbb{Z}<i>\rightarrow \mathbb{Z}_{2}</i> 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 don't know how to find the kernel, I know that by def ker\varphi=\{z\in\mathbb{Z}<i>; \varphi(z)=[0]_{2}\}</i>
please help :D

 

[micromass]

Homework Statement

let \varphi:\mathbb{Z}<i>\rightarrow \mathbb{Z}_{2}</i> 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 don't know how to find the kernel, I know that by def ker\varphi=\{z\in\mathbb{Z}<i>; \varphi(z)=[0]_{2}\}</i>
please help :D

Yes, so take a+bi \in \ker(\varphi. Then \varphi(a+bi)=0. Now just write things out using the definition of \varphi.