TITLE: Isomorphism $k[x,y]/(y-x^2)$ onto $k[x]$
QUESTION [6 upvotes]: Perhaps this is very easy but I would like to get a proof of isomorphism $k[x,y]/(y-x^2) \cong k[x]$.

REPLY [3 votes]: Can't this also be seen geometrically?  This may or may not be of interest to the OP, but I think it's worth posting. The only downside is that I think I need $k$ algebraically closed and of characteristic $0$. 
Recall that coordinate rings of affine varieties are isomorphic iff the varieties they cut out are. Next, notice that $k[x,y]/ \langle y-x^2 \rangle$ is the coordinate ring of the usual parabola centered at the origin in $k^2$ and $k[x] \simeq k[x]/\langle 0 \rangle \simeq k[x,y]/\langle y \rangle$ is  the coordinate ring of a copy of $k$. We can view this copy of $k$ as embedded into the plane, in which case the varieties are visually isomorphic, but the correspondence is given explicitly by $x \mapsto (x,x^2)$ or equivalently $(x,y^2) \mapsto x$, as others have noticed in different language.