# The number of elements in Gaussian Factor ring

• May 16th 2009, 07:21 AM
leonardo1000
The number of elements in Gaussian Factor ring
Hi Guys ;

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

(Itwasntme)
• May 16th 2009, 08:37 AM
Swlabr
Quote:

Originally Posted by leonardo1000
Hi Guys ;

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

(Itwasntme)

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 $\displaystyle |\mathbb Z_5[i]|/|<1+i>| = 25/|<1+i>|$.
• May 16th 2009, 01:33 PM
NonCommAlg
Quote:

Originally Posted by leonardo1000
Hi Guys ;

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

(Itwasntme)

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