How does one find the quotient in a polynomial ring in 2 and more variables?

Question

Given the polynomial ring in 2 variables, how do we prove the following: $k[x,y]/(y-x^2) \cong k[x,x^2]$ ? In one variable I can follow the argument. Also in case the ideal we take the quotient with is generated by monomials for which we can perform a polynomial division. Thanks for your comment or explanation.

Answer 1

Intuitively you 'divide out' any factors of $(y-x^2)$ just as you would with modular arithmetic. In the same way (if we denote the the reduction Mod $(y-x^2)$ by a bar) we have that $\overline{y-x^2} = \overline{0}$ so $\overline{y} = \overline{x^2}$.

Now to prove that $k[x,y]/(y-x^2) \cong k[x,x^2]$ we need the first Isomorphism theorem which states that if $\xi:R \to S$ is a ring homomorphism, then $R/\ker{\xi} \cong \xi(R)$.

We can construct this homomorphism as follows: \begin{align*} \xi: k[x,y] &\to k[x,x^2]\\ (x,y) &\mapsto (x,x^2) \end{align*}

(this is an evaluation map).
Now you only need to prove that the kernel is the ideal generated by $y-x^2$