Why are the polynomial rings $k[x]$ and $k[x,y]/\langle y-x^2 \rangle$ isomorphic? This is related to an exercise in Hartshorne's book which people have previously asked about, but I feel that this particular point hasn't been answered in a very simple way.
Let $k$ be an algebraically closed field. Let $k[x]$ be the ring of polynomials of one variable over $k$ and let $k[x,y]$ be the ring of polynomials of two variables over $k$. How can we see the isomorphism
$$
k[x] \simeq k[x,y]/\langle y-x^2 \rangle,
$$
where $\langle y-x^2 \rangle$ is the ideal in $k[x,y]$ generated by $y-x^2$?

Define 
$$\phi: k[x,y]\to k[x],\;x\to x,\;\;y\to x^2,\text{ and expand accordingly}$$
or if you prefer: $\phi f(x,y):=f(x,x^2)$. Now check stuff.
