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I have some general idea of going about this. I'm trying to come up with a suitable mapping. I've come up with $\phi\colon a + \langle x^2 + 1\rangle\longmapsto a$ where $a$ is an irreducible element in $\mathbb{Z}[i]$ but this mapping has some issues. For example take $5x + x^2 + 1$, which is $5x \pmod{x^2+1}$. $5x$ does not belong to $\mathbb{Z}[i]$. So my question is, is there a better mapping?

I know for a fact that if you add any integer call it $b$ to $\langle x^2 + 1\rangle$, you will get $b$ back which belongs to $\mathbb{Z}[i]$. I'm stuck when $b$ is not an integer.

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    I don't understand your definition of $\phi$.2011-05-14
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    How do you define Z[i]?2011-05-14
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    Its the ring of Gaussian integers: http://en.wikipedia.org/wiki/Gaussian_integer2011-05-14
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    @Person: yes, and how do you define the ring of Gaussian integers? Some people define it to be $\mathbb{Z}[x]/(x^2 + 1)$. I still don't understand your definition of $\phi$. What is $a'$?2011-05-14
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    @Person: Your proposed mapping does not even make sense. The elements of $\mathbb{Z}[x]/\langle x^2+1\rangle$ are represented by **polynomials** with integer coefficients. The elements of $\mathbb{Z}[i]$ are not polynomials.2011-05-14
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    @Person, but how do you define it?2011-05-14
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    @Person, see my answer for why I am asking "how do you defined Z[i]".2011-05-14
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    Couldn't you simply map $x$ to $i$?2011-05-14
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    Since $\mathbb Z[i]$ is defined as $\mathbb{Z}[x]/\langle x^2 + 1\rangle$ they are equal and hence the identity function is an isomorphism.2011-05-14
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    No definition of Z[i] is given, the question is meaningless.2011-05-14
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    @Pearson, I know what the Gaussian integers are, I'm asking how *you* defined Z[i].2011-05-14
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    You can't prove theorem without definition.2011-05-14
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    Voting to close as **not a real question**2011-05-14
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    @quanta: There are many standard ways of defining $\mathbb{Z}[i]$. Defining it as a quotient is just one of them. You can also define it as the smallest subring of $\mathbb{C}$ that contains $\mathbb{Z}$ and $i$, which is a pretty standard and which happens to agree with the notation $\mathbb{Z}[i]$.2011-05-14
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    Dear Person, This is one of several algebra questions you have asked in fairly rapid succession. One previous question was about the *meaning* of the notation $\langle x,3\rangle$. Before that, you asked a question about $\langle 2,x\rangle$ in which you included a precise *definition* of the notation in the question. You declined to explain why you were asking a new question about notation that you had already defined yourself in an earlier question. Without further explanation, it makes me wonder if you are somewhat lost in the algebra course that you are taking. Have you tried ...2011-05-14
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    ... discussing some of your questions directly with your professor, your TA, or other students in the class? Regards,2011-05-14

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