Let $k$ be algebraically closed. Show that $\mathbb{A}_k^1$ is not isomorphic to $C:= \{(x,y) \in \mathbb{A}_k^2 :x^2-y^2=1\}$.
I know that this is equivalent to showing that $k[t]$ and $k[x,y]/(x^2-y^2-1)$ are not isomorphic as $k$-algebras, but I don't know how to proceed.
$k[x]$ is a UFD, but not is $k[x,y]/(x^2-y^2-1)$ since you have $y.y=(x+1)(x-1)$.