1. ## 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.
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 $ab$.
to find a polynomial that has $a+b$ as a root, find the reduced Grobner basis for $.$ 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 $.$