Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by $$ f(x,y)=x^4-3xy+y^2,$$

$$ g(x,y)=x^5-4xy+3xy^2.$$

Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.

Is $x,x^2,x^3,x^4,x^5,x^6,x^7,x^8, x^9, x^{10}, x^{11}$ lies in $I$ respectively?

`KK.<x,y>=QQ[];I=Ideal([x^4-3*x*y+y^2,x^5-4*x*y+3*x*y^2]);x^2 in I`

$\endgroup$ – joro Apr 23 '15 at 9:48