# Gaussian integers.

• Apr 5th 2011, 06:01 PM
Deveno
Gaussian integers.
How can I prove that there is no isomorphism from Z[i]/<a+bi> to Zn, for any n, if (a,b) ≠1?
• Apr 5th 2011, 06:41 PM
tonio
Quote:

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

If by Zn you meant \$\displaystyle \mathbb{Z}_n:=\mathbb{Z}/n\mathbb{Z}\$ , then you can't since \$\displaystyle \mathbb{Z}[i]/<a+bi>\cong \mathbb{Z}_{a^2+b^2}\$

Tonio
• Apr 5th 2011, 07:11 PM
Deveno
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...
• Apr 6th 2011, 01:29 AM
tonio
Quote:

Originally Posted by Deveno
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...

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
• Apr 6th 2011, 02:17 AM
Deveno