How can I prove that there is no isomorphism from Z[i]/<a+bi> to Zn, for any n, if (a,b) ≠1?

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- April 5th 2011, 07:01 PMDevenoGaussian integers.
How can I prove that there is no isomorphism from Z[i]/<a+bi> to Zn, for any n, if (a,b) ≠1?

- April 5th 2011, 07:41 PMtonio
- April 5th 2011, 08:11 PMDeveno
is this true even if (a,b) ≠ 1? if (a,b) = 1, yes, i know that it's isomorphic to Z/(a^2+b^2)Z, although i would like to see a cleaner version of the isomorphism than what i came up with.

but what if a+bi = d(k+mi)? for example, what is Z[i]/<4+6i>? really needing some help, here... - April 6th 2011, 02:29 AMtonio

Ok, I missed that coprimality condition, but the answer's still the same, though it must be fixed.

You__really__want to read http://home.wlu.edu/~dresdeng/papers/factorrings.pdf , in particular

corollary 3.

Tonio - April 6th 2011, 03:17 AMDeveno
thanks, that is helpful.