Let

, where

is an algebraically closed field. The goal is to show that some ring obtained by inverting certain elements of

is isomorphic to a polynomial ring, again with certain elements inverted. This will show that the quotient field of

is isomorphic to the field of rational functions in some (should be 2) variables (which is the ACTUAL goal).

My first thought was to consider the ring

, where the relationship

can be rewritten as

. Then do a change of variables, say

, and then solve for

. That relation should then let me define a map to some localization of

.

I could be doing it right, but the equations seem to get way too complicated to check whether or not I have an isomorphism. Anyone have an idea?