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Math Help - Irreducibility over Integers mod p

  1. #1
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    Irreducibility over Integers mod p

    Prove that x^3-9 is irreducible over the integers mod 31.

    Prove that x^3-9 is reducible over the integers mod 11.

    Any help in this would be greatly appreciated as I study for my final exam.
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  2. #2
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    Quote Originally Posted by apalmer3 View Post
    Prove that x^3-9 is irreducible over the integers mod 31

    Prove that x^3-9 is reducible over the integers mod 11.

    Any help in this would be greatly appreciated as I study for my final exam.
    It is sufficient here to prove the polynomials have no zeros.
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  3. #3
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    Okay! I see how that helps prove that it's irreducible over the integers mod 31... but how does that prove that it's reducible over integers mod 11?

    Thanks for the help!
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    Actually... I don't understand...

    For example:
    x^2+1 is reducible mod 5. It's equal to (x+2)(x+3). How can that be when (real) roots don't exist?
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    Quote Originally Posted by apalmer3 View Post
    but how does that prove that it's reducible over integers mod 11?
    Note 4 is a zero, thus, (x-4) is a factor of x^3 - 9.
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  6. #6
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    Quote Originally Posted by apalmer3 View Post
    Actually... I don't understand...

    For example:
    x^2+1 is reducible mod 5. It's equal to (x+2)(x+3). How can that be when (real) roots don't exist?
    That only works for polynomials over \mathbb{R}. You are in a different field, i.e. mod 5.
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  7. #7
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    Quote Originally Posted by ThePerfectHacker View Post
    That only works for polynomials over \mathbb{R}. You are in a different field, i.e. mod 5.

    Okay. But (x+2)(x+3) = x^2+5x+6

    In the integers mod 5, that equals x^2+0x+1 = x^2+1... see what I mean?
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