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Math Help - UFD

  1. #1
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    UFD

    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!
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by karlito03 View Post
    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 \phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}_p[x] (you are just quotienting out the ideal generated by p, p\mathbb{Z}).
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  3. #3
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    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)
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by karlito03 View Post
    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 A \Rightarrow B you prove not B \Rightarrow not A. These two things are equivalent.
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