The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of
[Dicks][1] or the proof of
[Mckay and Wang][2].

What can be said about the group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$? 

Of course, every $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ yields a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$, but there are more $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$, for example,
$x \mapsto x^{-1}, y \mapsto y$. 

Can one find all of them and characterize that group?

  


  [1]: http://www.raco.cat/index.php/PublicacionsSeccioMatematiques/article/viewFile/37473/37347
  [2]: http://www.sciencedirect.com/science/article/pii/0022404988901375