While studying Grobner basis, I realized that creating a basis from a given set of polynomials is not that hard, it is reduced to solving with Gauss Jordan a system of equations. What I don't understand is how to use them for other purposes. I have two examples here that I don't understand how to solve:
- Find an ideal in Q[x, y] whose reduced Grobner basis (in lexicographic order) has cardinality 5 and there are precisely 19 standard monomials.
- Using Grobner bases, find the minimal polynomials of $\sqrt[5]{6} + \sqrt[7]{8}$ and $\sqrt[5]{6} - \sqrt[7]{8}$.
How can I find the minimal polynomial using Grobner bases if I don't know for what polynomials to take the bases?