I have few questions about the following two exercises:
1. Compute the remainder on dividing $x^2y=f$ by $G=\{ x^2+xy+z^2, y^2+2z^2 \}$. You may use the fact that $G$ is a Gröbner basis for the ideal $\left< x^2+xy+z^2, y^2+2z^2 \right>$ with respect to the lex order and $x>y>z$.
So I did the division and I got $2z^2x-z^2y$ for the remainder. However I do not understand the hint and what it implies and I guess I should use it to answer to the question.
2. Is $x^2+1$ in the ideal $\left< x^2+xy+z^2, y^2+2z^2 \right>$ which is included in $\mathbb{Q}[x,y,z]$?
I have the feeling these two questions are closely related and I will say the answer to 2 is no since the remainder of the previous question is not zero.
But I believe I am pretty wrong for both questions.
Thanks for your help
A remainder on dividing a polynomial f by a set G of polynomials is not necessarily unique, it becomes unique when G is a Groebner basis for . That's why in the first question they have specified "the" remainder. You are correct to say that both questions are linked, because in order to solve the second question, you need first to compute a Groebner basis G for your ideal and then divide f by G. If the remainder if zero then f belongs in . I hope this help.