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.


In the lines of the accepted answer, one can show that the global dimension of $k[x,x^{-1}]$ is $1$, so that if it is isomorphic to a polynomial ring, it must be isomorphic to one of the form $D[y]$ with $D$ a semisimple commutative domain, which is thus a field. 
