I was doing an exercise, and to finish it, I need to find something, and I conclude that is just this group of automorphism. The questions is, I can say more about it? ( There has some form, etc..) ? So the question is: Given a field k , Consider the polynomial ring $ k[x_1,x_2,...x_n]$. It´s possible to find all the automorphism of this ring, that fix the field k?
Let $X = (x_1, x_2, \ldots , x_n)$. Automorphisms are polynomial substitutions $f(X) \to f(P(X))$ where $P$ has a polynomial inverse.
Structure theory of the automorphism group is complicated for $n \geq 3$. A recent breakthrough was the proof that not all of the group is generated by substitutions $x_i \to ax_i +b$ where $a$ is a constant and $b$ is a polynomial in the other variables.
For $n=2$ these substitutions (called tame automorphisms) generate the full automorphism group and the structure of the group is explained in Cohn's book on free rings. The structure of the tame subgroup for $n=3$ is determined in
This is all very closely connected to the Jacobian conjecture.