1. 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.

2. Prove that Z_p is a unique factorization domain for prime p.

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.

3. ???

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?

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

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!)

6. ....

i can't use that assumption. i have to prove it as if i don't know anything above.

7. This whole question is alewady under discussion here: http://www.mathhelpforum.com/math-he...need-help.html