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Gaussian integers, ring homomorphism and kernel

  • Thread starter rayman123
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  • #1
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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}[/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 dont 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 :D
 

Answers and Replies

  • #2
22,097
3,280

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}[/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]....


Maybe bd=-bd (mod 2) ??

I dont 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 :D


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

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