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Let $K$ be a field of characteristic $0$ and let $\sigma$ be the element of $Aut(K(x)/K)$ defined by $(\sigma f)(x) = f(x + 1)$. Show that $U = \langle\sigma\rangle$ is an infinite cyclic group; determine its fixed field $F(U)$ and the degree $[K(x) : F(U)]$.

$F(U) = \{f \in K(X) \mid \sigma f = f \ \forall \ \sigma \in U\} = \{f \in K(X) \mid f(x) = f(x+1) = f(x+2)= \cdots \}$

Thus $f \in F(U)$ must be a constant function. i.e. $K$ is the fixed field of $U$.

Is the argument of my problem correct?

Artin's Theorem: Let $(L : K)$ he a field extension. If $U$ is a finite subgroup of $Aut(L/K)$, then $[L : F(U)] = |U|$.

But here $U$ is not finite. Is there any other corr. theorem to find the degree $[K(x) : F(U)]$?

$f\in K$ is of the form $p(x)/q(x)$ where they are co-prime. The equality $p(t)/q(t)=p(t+1)/q(t+1)$ implies that $p(t)$ and $p(t+1)$ have the same zeroes in $k$, and the same for $q(t)$ and $q(t+1)$. This is impossible, unless they are constant.

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    The automorphism is defined by $\sigma(x) = x+1$. If $char(K) = 0$ then $\sigma^n(x) = x+n \ne x$ so $\langle \sigma \rangle$ is isomorphic to $\mathbb{Z}$. And for the fixed field, you can look at the roots and poles of $\frac{f(x)}{g(x)} \in \overline{K}(x)$ and see how $\sigma$ acts on them – 2017-01-10
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    Yes i have got the fact that U is infinite cyclic – 2017-01-10
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    You need an argument for showing that $f(x+n) \ne f(x)$. I say you can look at the roots and poles in the algebraic closure (similar to the case $K = \mathbb{C}$) – 2017-01-10
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    Your argument for $F(U) = K$ is pretty much completely unjustified. You should provide some reasoning that $f$ must be constant (will probably rely on the fact that a nonzero polynomial has finitely many roots). Also, if $F(U) = K$, then the degree of $K(x)/K$ shouldn't be too hard to compute by hand. – 2017-01-10
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    And the "[transcendence degree](https://en.wikipedia.org/wiki/Transcendence_degree)" is not the same as the "[degree of a field extension](https://en.wikipedia.org/wiki/Degree_of_a_field_extension)". So you should really get and read a course, instead of asking many such questions. – 2017-01-10

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