# Unique Factorization Domains

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• Dec 17th 2009, 08:14 AM
pleasehelpme1
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, 09:25 PM
NonCommAlg
Prove that Z_p is a unique factorization domain for prime p.
Quote:

Originally Posted by pleasehelpme1
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.

$\mathbb{Z}_p$ is a field and so $\mathbb{Z}_p[x]$ is a PID. now try to recall that every PID is a UFD.
• Dec 18th 2009, 06:24 AM
pleasehelpme1
???
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, 12:18 PM
tonio
Quote:

Originally Posted by pleasehelpme1
Prove that every irreducible is prime in Z_p where p is prime.

You are aware of the fact that $\mathbb{Z}_p$ is a FIELD for any prime $p$ ?? 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, 01:29 PM
NonCommAlg
Quote:

Originally Posted by pleasehelpme1
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, 02:51 PM
pleasehelpme1
....
i can't use that assumption. i have to prove it as if i don't know anything above.
• Dec 18th 2009, 04:52 PM
mr fantastic
This whole question is alewady under discussion here: http://www.mathhelpforum.com/math-he...need-help.html

Thread closed.