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Math Help - Grobner basis technique

  1. #1
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    Grobner basis technique

    Let R and S be Noetherian integral domain with R \subset S. Suppose that a,b \in S are roots of the monic polynomials
    x^2+c_1x+c_0, x^2+d_1x+d_0 \in R[x] respectively.
    Using Grobner basis techniques, exhibit a monic polynomial that has a+b as a root. Do the same for ab.

    I know method for computing Grobner basis but I don't know how I use Grobner basis techniques for this problem.
    Please, give me some hint.
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  2. #2
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    Quote Originally Posted by Byun View Post
    Let R and S be Noetherian integral domain with R \subset S. Suppose that a,b \in S are roots of the monic polynomials
    x^2+c_1x+c_0, x^2+d_1x+d_0 \in R[x] respectively.
    Using Grobner basis techniques, exhibit a monic polynomial that has a+b as a root. Do the same for ab.

    I know method for computing Grobner basis but I don't know how I use Grobner basis techniques for this problem.
    Please, give me some hint.
    to find a polynomial that has a+b as a root, find the reduced Grobner basis for <x^2 + c_1x+c_0, \ y^2 + d_1y + d_0, \ x+y-z>. then the element of your basis which is in R[z] will have a+b

    as a root. to find a polynomial that has ab as a root, do the same as above but this time for <x^2 + c_1x + c_0, \ y^2 + d_1y + d_0, \ xy - z >.
    Last edited by NonCommAlg; January 12th 2009 at 08:37 PM.
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