# integral domain?

• Oct 23rd 2008, 07:18 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, 07:35 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, 08:36 AM
ThePerfectHacker
Quote:

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

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

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

Therefore \$\displaystyle n\$ must be prime.
• Oct 23rd 2008, 09:03 AM
kbartlett
Lol im so stuipd, it was already in my lecture notes anyway.

But thanks