1. ## Grobner basis technique

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

I know method for computing Grobner basis but I don't know how I use Grobner basis techniques for this problem.
Let $\displaystyle R$ and $\displaystyle S$ be Noetherian integral domain with $\displaystyle R \subset S$. Suppose that $\displaystyle a,b \in S$ are roots of the monic polynomials
$\displaystyle x^2+c_1x+c_0, x^2+d_1x+d_0 \in R[x]$ respectively.
Using Grobner basis techniques, exhibit a monic polynomial that has $\displaystyle a+b$ as a root. Do the same for $\displaystyle ab$.
to find a polynomial that has $\displaystyle a+b$ as a root, find the reduced Grobner basis for $\displaystyle <x^2 + c_1x+c_0, \ y^2 + d_1y + d_0, \ x+y-z>.$ then the element of your basis which is in $\displaystyle R[z]$ will have $\displaystyle a+b$
as a root. to find a polynomial that has $\displaystyle ab$ as a root, do the same as above but this time for $\displaystyle <x^2 + c_1x + c_0, \ y^2 + d_1y + d_0, \ xy - z >.$