I am trying to understand the Milnor number introduced on Wikipedia. The second example is for a polynomial $f(x,y) = x^3 +x y^2$ with derivatives $f_{x}=3 x^2 +y^2$ and $f_{y} = 2 x y$. In the example, it is written that the basis of the quotient space reads $\{ 1,x,y, x^2 \}$, resulting in the Milnor number $\mu=4$.
Could you explain why $x^{2}$ is in the set? Based on the $f_{x}$ polynomial, I thought the basis should not have $x^2$ and $y^{2}$.
