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Math Help - Greatest common divisor of two polynomials

  1. #1
    Super Member Bernhard's Avatar
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    Greatest common divisor of two polynomials

    I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8

    ================================================== ==================================

    Determine the greatest common divisor of  a(x) = x^3 - 2 and  b(x) = x + 1 in  \mathbb{Q} [x]

    and write it as a linear combination (in  \mathbb{Q} [x] ) of a(x) and b(x).

    ================================================== ===================================

    In working on this I applied the Division Algorithm to a(x) and b(x) resulting in


     x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3)

    then

     (x + 1) = (1/3 x + 1/3) + 0


    Last non-zero remainder is -3

    Therefore, gcd is -3

    BUT!

    This does not seem to be correct because -3 does not divide either a(x) and b(x)

    Can someone please help?

    Peter
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  2. #2
    Super Member
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    Re: Greatest common divisor of two polynomials

    Hi,
    I think your plan of attack is exactly correct. Remember you are working in Q[x], not Z[x]. So in Q[x], -3 does divide any polynomial; e.g. x^3-2 =-3(-1/3x^3-2/3). That is -3 is a unit in Q[x].
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  3. #3
    Junior Member Mathhead200's Avatar
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    Re: Greatest common divisor of two polynomials

    Using the rational roots theorem, you can check that a(x) has no rational roots. I pretty sure this means it has no (non-unit) factors in \mathbb{Q}\[x\]. This would make a(x) \in \mathbb{Q}\[x\] irreducible. Therefor, I believe gcd( a(x), b(x) ) = u, any unit in \mathbb{Q}\[x\], i.e. any (non-zero) constant polynomial. Since there ordering on \mathbb{Q}\[x\] is only concerned with the polynomials degree, I don't think any constant polynomial would be called "greater" then any other.
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