Exercise 4, § 14

Let K be a commutative field, $\sigma$ an automorphism of K distinct from the identity, and D the field $K((X))_{\sigma}$ (VIII, p. 19, Exercise 21, e)). Let $\tau$ be an automorphism of K that commutes with $\sigma$. Prove that there exists a unique automorphism of D extending $\tau$ and fixing X. Deduce an example of a field automorphism that fixes the center and is not inner.