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Math Help - irreducible polynomial

  1. #1
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    irreducible polynomial

    I'm just now trying to do another related exam question:

    Show that g = X^3 + 2X - 1 is irreducible over rationals.

    If x = b/c, b must be a factor of -1, c must be a factor of 1.

    Possible factors of -1: +1, -1
    Possible factors of 1: +1, -1

    List of possible rational roots:

    Direct checking shows that none of these are a root and hence g has no rational root.

    Therefore since there are no rational roots g cannot be irreducible over the rationals.


    Is this the right approach for this question?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by hunkydory19 View Post
    I'm just now trying to do another related exam question:

    Show that g = X^3 + 2X - 1 is irreducible over rationals.

    If x = b/c, b must be a factor of -1, c must be a factor of 1.

    Possible factors of -1: +1, -1
    Possible factors of 1: +1, -1

    List of possible rational roots:

    Direct checking shows that none of these are a root and hence g has no rational root.


    Therefore since there are no rational roots g cannot be irreducible over the rationals.


    Is this the right approach for this question?

    Thanks in advance!
    This seems like the right approach to me. In fact, I don't know how else you would do it. Your conclusion, however, should be that g is irreducible. I'm sure you just made a typo.
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  3. #3
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    Quote Originally Posted by hunkydory19 View Post
    I'm just now trying to do another related exam question:

    Show that g = X^3 + 2X - 1 is irreducible over rationals.

    If x = b/c, b must be a factor of -1, c must be a factor of 1.

    Possible factors of -1: +1, -1
    Possible factors of 1: +1, -1

    List of possible rational roots:

    Direct checking shows that none of these are a root and hence g has no rational root.

    Therefore since there are no rational roots g cannot be irreducible over the rationals.


    Is this the right approach for this question?

    Thanks in advance!
    Yes. You need to remember a polynomial is reducible over Q if and only if it is reducible over Z. This is why you are allowed to restrict your attention to
    integers.
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  4. #4
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    Well someone else told me that I should be using Gauss's lemma, so I wasn't sure if it was OK to use this method, but since it is I'll definitely be using this as it's much easier!

    Thank you for your help!
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