# Thread: The number of elements in Gaussian Factor ring

1. ## The number of elements in Gaussian Factor ring

Hi Guys ;

how many elements are in $\mathbb Z_5[i]/<1+i>$ ?

2. Originally Posted by leonardo1000
Hi Guys ;

how many elements are in $\mathbb Z_5[i]/<1+i>$ ?

How many elements are in $\mathbb Z_5[i]$? What about $<1+i>$? Clearly, $\mathbb Z_5[i]$ has 25 elements, so $<1+i>$ 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 $|\mathbb Z_5[i]|/|<1+i>| = 25/|<1+i>|$.

3. Originally Posted by leonardo1000
Hi Guys ;

how many elements are in $\mathbb Z_5[i]/<1+i>$ ?

one element only! the reason is that in $\mathbb{Z}_5[i]$ we have $1=(3+2i)(1+i) \in <1+i>.$ thus $<1+i>=\mathbb{Z}_5[i].$