Number of generators of an ideal in $k[x,y]$

Question

Let $k$ be any field. We consider the ideal $(x^2, xy, y^2)$ in $k[x,y]$. I need to prove that this ideal can not be generated by two elements.

I tried this by considering the contrapositive, but can't see anything useful. I guess some results from dimension theory will be required. Any help is highly appreciated.

Answer 1

Set $I=(x^2, xy, y^2)$ and $m=(x,y).$
If $I=(f,g)$, then k-vector, $I/mI$, space has dimension at most $2$. But $x^2+mI, xy+mI, y^2+mI$ are linearly independent in $I/mI$; since every element of $mI$ has degree at least $3.$