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I'm working through Q3 of this paper as part of my revision for forthcoming exams.

I'm struggling with how to tackle the question:

Show that there is a homomorphism from the ring of Gaussian integers $Z[i]$ onto $F$; what is its kernel?

Can anyone help identify this homomorphism and find its kernel?

Any help would be appreciated. Thanks!

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your link isn't working correctly for me. What is F? –  Mr.Guy Jan 12 '13 at 1:07
    
I've corrected the link for everyone. –  Mathmo Jan 12 '13 at 1:10
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It's generally a good to present a complete problem in your questions. Putting crucial information in a link might save you a few seconds, but wastes everybody else's time, and will invalidate your question in the future when the link gets broken. (having the link is probably fine, though, if adequate detail is present in your question) –  Hurkyl Jan 12 '13 at 1:18
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1 Answer

up vote 4 down vote accepted

Your homomorphism will involve the relationship in your F. F = $Z_7$[$x$]/($x^2+1$). So $x^2 = -1$. So all elements in F are of the form: a+bx where a,b are in $Z_7$ and with the relationship that $x^2 = -1$ So if we take the Gaussian integers which are of the form a + bi where a, b exist in the integers and $i^2 = -1$. We can send a and b to $Z_7$ by reducing them modular 7 and we can send i to x. Then all of our relationship will hold and our kernel will be everything that goes to 0. Since i goes to x the coefficient of i must be 0. So this results in the kernel being $7*Z$.

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Try to write your answers with LaTeX to make them readable. Go the FAQ section for directions. +1 –  DonAntonio Jan 12 '13 at 1:36
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