Hi Guys ;
how many elements are in $\displaystyle \mathbb Z_5[i]/<1+i>$ ?
How many elements are in $\displaystyle \mathbb Z_5[i]$? What about $\displaystyle <1+i>$? Clearly, $\displaystyle \mathbb Z_5[i]$ has 25 elements, so $\displaystyle <1+i>$ has either 1, 5 or 25 elements. It's not got only 1 element, obviously (as it contains 0). I'll leave the rest of that bit to you...and the solution to your problem is simply $\displaystyle |\mathbb Z_5[i]|/|<1+i>| = 25/|<1+i>|$.