# Proving a pair of integers are not prime in a particular number system

• May 16th 2011, 06:25 PM
burdo
Proving a pair of integers are not prime in a particular number system
Basically i've been given a number system G defined as follows, numbers in G are ordered pairs of integers i.e. (a,b)\inG if a\inZ and b\inZ. Z is the set of all integers. Addition and multiplication are defined for G as follows

(a,b) + (c,d) = (a+c,b+d)

(a,b) x (c,d) = (ac-bd,ad+bc)

I have to show that (2,0) is not prime in G even though 2 is prime in Z. I'm then asked to find two factors of (2,0) that aren't units.

I've been stuck on this question for ages now and i've had no luck with it. My lecture notes havn't helped too much. Any help would be really appreciated.

Thanks, Mark
• May 16th 2011, 07:54 PM
Drexel28
Quote:

Originally Posted by burdo
Basically i've been given a number system G defined as follows, numbers in G are ordered pairs of integers i.e. (a,b)\inG if a\inZ and b\inZ. Z is the set of all integers. Addition and multiplication are defined for G as follows

(a,b) + (c,d) = (a+c,b+d)

(a,b) x (c,d) = (ac-bd,ad+bc)

I have to show that (2,0) is not prime in G even though 2 is prime in Z. I'm then asked to find two factors of (2,0) that aren't units.

I've been stuck on this question for ages now and i've had no luck with it. My lecture notes havn't helped too much. Any help would be really appreciated.

Thanks, Mark

This is a question about algebra. and secretly $G$ is just the Gaussian integers $\mathbb{Z}[i]=\left\{a+bi:a,b\in\mathbb{Z}\right\}$ and then you can quickly check that $2=1-(-1)=1-i^2=(1-i)(1+i)$ yet the only units in $\mathbb{Z}[i]$ (as you can check) are $1,-1,i,-i$.