# Unique Factorization Domains

• Dec 17th 2009, 07:14 AM
Unique Factorization Domains
Let p be a prime integer.

Prove that Z_p[x] is a unique factorization domain.

I need a legit proof to this if possible.
• Dec 17th 2009, 08:25 PM
NonCommAlg
Prove that Z_p is a unique factorization domain for prime p.
Quote:

Let p be a prime integer.

Prove that Z_p[x] is a unique factorization domain.

I need a legit proof to this if possible.

\$\displaystyle \mathbb{Z}_p\$ is a field and so \$\displaystyle \mathbb{Z}_p[x]\$ is a PID. now try to recall that every PID is a UFD.
• Dec 18th 2009, 05:24 AM
???
why is this always the answer i get... you can just assume things like Z_p being a field. I don't see how that proves anything?
• Dec 18th 2009, 11:18 AM
tonio
Quote:

Prove that every irreducible is prime in Z_p where p is prime.

You are aware of the fact that \$\displaystyle \mathbb{Z}_p\$ is a FIELD for any prime \$\displaystyle p\$ ?? Because if you are then your question is trivial, and if you aren't then perhaps try to prove it: it isn't hard.

Tonio
• Dec 18th 2009, 12:29 PM
NonCommAlg
Quote:

Let p prime.

Prove that Z_p is a unique factorization domain.

a field is a PID and a PID is a UFD. (take a look at your lecture notes!)
• Dec 18th 2009, 01:51 PM