Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable. 
Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that
  $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.


We have $k[x,\frac{1}{x}]$ as a ring. Form a polynomial ring $k[x,\frac{1}{x}][y]$ with y as the variable. Construct an evaluation map $\phi$ $$\phi:k[x,\frac{1}{x}][y] \to k[x,\frac{1}{x}], \text{with }x\to x, \frac{1}{x} \to \frac{1}{x}, y \to \frac{1}{x}$$ then $\phi$ is obviously a ring morphism. Now by factor theorem, if $f\in k[x,\frac{1}{x}][y], f\in \ker\phi$, then $f(1/x) = 0$, which gives $(y-1/x)|f$. Note that 1/x is a taken as a constant here.
Now if $f\in k[x][y] \subset k[x,\frac{1}{x}][y] $, then $f = g(y-1/x), g \in k[x,\frac{1}{x}][y]$. Now you only need to show that g is in $k[x][y]$ and then we've shown $k[x][y]/(xy-1) \cong k[x,1/x]$, now check for the accepted answer by user26857.
