UFD

May 2010
12
0
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!
 
May 2009
1,176
412
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 \(\displaystyle p\), \(\displaystyle p\mathbb{Z}\)).
 
May 2010
12
0
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)
 
May 2009
1,176
412
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.