Let $\mathcal{O}_{k^2,(0,0)}$ be the local ring at origin, i.e. $k[x,y]_{(x,y)}$.
I want to show that $\dim_k \mathcal{O}_{k^2,(0,0)}/(y-x^2,x^3)=3$.
My rough argument is the following, but I feel that this is somewhat not rigorous:
Suppose $\alpha=\dfrac {f(x,y)}{g(x,y)},\ g(0,0)\neq 0$ is an element of $\mathcal{O}_{k^2,(0,0)}/(y-x^2,x^3)$. Then substituting $y=x^2, > x^3=0$, we can make $\alpha = \dfrac{f_0+f_1x+f_2x^2}{g_0+g_1x+g_2x^2}$. Also by multiplying an appropriate polynomial $1+ax+bx^2$ to denominator and numerator, we can make the denominator to be constant so that $\alpha= A+Bx^2+Cx^2$. So the dimension is 3.
Can I make it mathematically rigorous? or is it the best?