I need to prove that $X=Z(xy-1)\subset \mathbb{A}_k^2$ is not isomorphic to $\mathbb{A}^1_k$.
I solved an exercise where I proved that, for instance, some $X\subset \mathbb{A}_k^3$ is isomorphic to $\mathbb{A}^2_k$, by constructing a morphism and showing that it is bijection (is the idea correct?).
However, I am stucked to show that is not isomorphic.
Well, I thought that $\mathbb{A}^1_k$ being a field and $X$ not (because $(0,0)\not\in X$), so they are not isomorphic.
However, this is as I've done with homomorphisms (rings, etc.), not morphisms between algebraic sets. I do not know if I can consider such thing as be or not a field.
Many thanks in advance!
You need to show the coordinate rings are non-isomorphic as $k$-algebras. The coordinate ring of $\Bbb A_k^1$ is just $k[t]$. That of $X$ is $$\frac{k[x,y]}{(xy-1)}\cong k[x,x^{-1}].$$ You need to show this ring isn't a polynomial ring. What say are the units of $k[x,x^{-1}]$?