Hi guys, I need some help with the following exercise

Consider the Guassian intergers. Let J be an ideal of Z[i]. Prove that the quotient Z[i]/J is a field of 9 elements.

J=

I know the quotient is a field but I donīt know why is has 9 elements.

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- May 21st 2011, 04:09 PMorbitQuotient ring and field of 9 elements
Hi guys, I need some help with the following exercise

Consider the Guassian intergers. Let J be an ideal of Z[i]. Prove that the quotient Z[i]/J is a field of 9 elements.

J=

I know the quotient is a field but I donīt know why is has 9 elements. - May 21st 2011, 07:34 PMtonio
- May 21st 2011, 07:43 PMorbit
- May 21st 2011, 07:59 PMDrexel28
- May 21st 2011, 08:14 PMorbit
sorry, I dont get it.

- May 21st 2011, 09:59 PMDrexel28
- May 21st 2011, 10:15 PMorbit
The part I dont understand is why a,b belongs to {0,1,2}....

- May 21st 2011, 10:45 PMDeveno
suppose a = k+3m, b = n+3r.

then doesn't a+bi + J = k+ni + J? - May 21st 2011, 11:18 PMorbit
- May 21st 2011, 11:29 PMDeveno
what you know is that the coset a+bi + J is the same as a(mod 3) + b(mod 3)i + J. that is a,b = 0,1 or 2 (mod 3). this means we get exactly 9 (distinct) elements:

0+0i + J, 0+i + J, 1+0i + J, 1+i + J, 0+2i + J, 2+0i + J, 1+2i + J, 2+i + J, 2+2i + J. - May 22nd 2011, 12:00 AMorbit
- May 22nd 2011, 03:05 AMtonio
- May 22nd 2011, 07:50 AMDeveno
as Z[i] is an integral domain, to show J is maximal it suffices to show that J is:____?