Fixed Field of Automorphisms of $k(x)$

abstract-algebra field-theory galois-theory

Hint:

  • The sum $$t=x+\frac1{1-x}+\frac{x-1}x=\frac{x^3-3x+1}{x^2-x}$$ is stable under the action of your group of transformations.
  • Can you show that $[k(x):k(t)]=3$?

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