Grobner basis technique

**Byun** · January 12th 2009, 05:30 PM

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.

**NonCommAlg** · January 12th 2009, 08:22 PM

Originally Posted by Byun

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 $<x^2 + c_1x + c_0, \ y^2 + d_1y + d_0, \ xy - z >.$

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