# UFD

#### karlito03

I have been struggling with this can someone help Please?

a.Prove that if a polynomial is irreducible in Z_p[x], then it
is irreducible in Z[x].

b.Prove that if a polynomial factors in Z[x], then it factors in
Z_p[x] for some prime p.

Thank u!

#### Swlabr

I have been struggling with this can someone help Please?

a.Prove that if a polynomial is irreducible in Z_p[x], then it
is irreducible in Z[x].

b.Prove that if a polynomial factors in Z[x], then it factors in
Z_p[x] for some prime p.

Thank u!
HINT: Use the fact that there exists a homomorphism of rings $$\displaystyle \phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}_p[x]$$ (you are just quotienting out the ideal generated by $$\displaystyle p$$, $$\displaystyle p\mathbb{Z}$$).

#### karlito03

Thank I'll try that. Isn't that just for question (b) only though? I don't see how I would I apply this to (a)

#### Swlabr

Thank I'll try that. Isn't that just for question (b) only though? I don't see how I would I apply this to (a)
Personally, I would do question (b), then extend it for all primes p. This is the contrapositive of (a)*, so you are done.

That is a bit roundabout though, so there may be an easier way...

*The contrapositive: Instead of proving $$\displaystyle A \Rightarrow B$$ you prove $$\displaystyle not B \Rightarrow not A$$. These two things are equivalent.