Calculating Grobner Bases

Question

In this question, $ℚ[x,y,z]$ is endowed with the lexicographic order with $x > y > z$. Set $u:= x^2 + 2yz^2$ and $v:= y^2 - 3xz$. Denote by $J$ the ideal of $ℚ[x,y,z]$ generated by $u$ and $v$.

1. Compute a reduced Gröbner basis of $J$ by using Buchberger's algorithm.
2. Does $f:= x^2 - xz + 4$ belong to $J$?
3. Prove that $V(J)$ is infinite.
4. Does $g:= (x + y + z)^{2014}$ belong to $J$?