- #1

rayman123

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## Homework Statement

let [tex]\varphi:\mathbb{Z}

*\rightarrow \mathbb{Z}_{2}[/tex] be the map for which [tex]\varphi(a+bi)=[a+b]_{2}[/tex]*

a)verify that [tex]\varphi[/tex] is a ring homomorphism and determine its kernel

b) find a Gaussian integer z=a+bi s.t [tex]ker\varphi=(a+bi)[/tex]

c)show that [tex]ker\varphi[/tex] is maximal ideal in [tex]\mathbb{Z}

a)verify that [tex]\varphi[/tex] is a ring homomorphism and determine its kernel

b) find a Gaussian integer z=a+bi s.t [tex]ker\varphi=(a+bi)[/tex]

c)show that [tex]ker\varphi[/tex] is maximal ideal in [tex]\mathbb{Z}

*[/tex]*

I started by showing that [tex]\varphi[/tex] preserves the ring operations

[tex]\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)[/tex]

and multiplication

[tex]\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[/tex]

but something is not right here because if I look at the right hand side, I should get

[tex]\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd[/tex]...

I don't know how to find the kernel, I know that by def [tex]ker\varphi=\{z\in\mathbb{Z}I started by showing that [tex]\varphi[/tex] preserves the ring operations

[tex]\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)[/tex]

and multiplication

[tex]\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[/tex]

but something is not right here because if I look at the right hand side, I should get

[tex]\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd[/tex]...

I don't know how to find the kernel, I know that by def [tex]ker\varphi=\{z\in\mathbb{Z}

*; \varphi(z)=[0]_{2}\}[/tex]*

please help :Dplease help :D