Is Aut(F) a field under point-wise addition and multiplication of functions?

Question

Does Aut(F), the set of automorphisms of a field F, form a field under point-wise addition and multiplication of functions?

Answer 1

Aut(F) is clearly a group, under function composition. It is not a field, however.

If f is an Automorphism of F, then so is -f.
But f + (-f) = 0, while the zero map is certainly not an automorphism. Thus Aut(F) is not closed under addition, so cannot be a field, or even a ring when using pointwise addition as an operation.

Answer 2

I do not believe it forms a field (it does however form a group). I do not have a proof that it does not form a field, but I dont think so