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Math Help - Irreducible????

  1. #1
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    Irreducible????

    Question:
    Determine if x^4 + x^2 + 1 is ireducible in Z3[x]. Factorize it if you can

    I think x^4 + x^2 + 1 can be factored as
    (x^2 +2)(x^2 +2)

    which would make it Irreducible but not sure?

    Please Help
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  2. #2
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    Quote Originally Posted by mathlg
    Question:
    Determine if x^4 + x^2 + 1 is ireducible in Z3[x]. Factorize it if you can

    I think x^4 + x^2 + 1 can be factored as
    (x^2 +2)(x^2 +2)

    which would make it Irreducible but not sure?

    Please Help
    Notice that,
    (x^2+2)(x^2+2)=x^4+4x^2+4
    Since we are in \mathbb{Z}_3[x] we have that,
    x^4+x^2+1.
    Which means it is reducible.

    Irreducible over a Field means a polynomial CANNOT be factored into non-constant polynomials as elements of that field. Which is not true here.
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  3. #3
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    OK I see what you are saying. I forgot about the Z3 [x] which makes it reducible
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