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Assume $\operatorname{char}K = 0$ and let $G$ be the subgroup of $\operatorname{Aut}_K K(x)$ that is generated by the automorphism induced by $x\mapsto x + 1_K.$ Then $G$ is an infinite cyclic group. Determine the fixed field $E$ of $G.$ What is $[K(x) : E]$?

I think that this exercise begin by using first part of Artin's lemma and get $K(x)$ is Galois over $E.$ I.e., $G < \operatorname{Aut}(K(x)/E)$, we get $E=G'=G'''=E''$ lemma V.2.6-(iv) of Hungerford.

But I don't know how to proceed. Can someone help me?

Stahl
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    An argument is buried in this old answer of mine. That question is nominally about the fixed field of a dihedral group containing your group as an index two subgroup. However, I think you can use that argument. – Jyrki Lahtonen Sep 13 '21 at 11:03