I am reading through the wikipedia page of Milnor number.
I am reading example 2 where they calculate the Milnor number of $f(x,y)=x^3+xy^2$. So if we calculate the partial derivatives we obtain $\frac{\partial f}{\partial x}=3x^2+y^2$ and $\frac{\partial f}{\partial y}=2xy$, so we have that $$\mu (f)=\text{dim}_{\mathbb{C}}\mathbb{C}[\![x,y]\!]/(3x^2+y^2,xy)$$
But if I look at this, it seems to me that we have a generating set $\{1,x,y,x^2\}$, but the basis should consist of infinitely many monomials, since we cannot write for example $x^n$ for $n>2$ as a linear combination of the above monomials. But wikipedia claims that $\mu(f)=4$.
My question: what am I missing/doing wrong?