# integral domain?

• Oct 23rd 2008, 08:18 AM
kbartlett
integral domain?
Im having problems workout out
For which n are the integers mod n an integral domain?

can any1 show me how to prove this
• Oct 23rd 2008, 08:35 AM
Jhevon
Quote:

Originally Posted by kbartlett
Im having problems workout out
For which n are the integers mod n an integral domain?

can any1 show me how to prove this

ok, lets start with the basics. do you know what an integral domain is?
• Oct 23rd 2008, 09:36 AM
ThePerfectHacker
Quote:

Originally Posted by kbartlett
Im having problems workout out
For which n are the integers mod n an integral domain?

For $\mathbb{Z}_n$ we require that if $[a]_n[b]_n = [0]_n$ implies $[a]_n=[0]_n$ or $[b]_n=[0]_n$.

Now if $n$ is not prime there is $1 < d < n$ so that $d|n$. Then $[d]_n [n/d]_n = [0]_n$ but neither $[d]_n,[n/d]_n$ are identity elements.

Therefore $n$ must be prime.
• Oct 23rd 2008, 10:03 AM
kbartlett
Lol im so stuipd, it was already in my lecture notes anyway.

But thanks