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We consider a field $K$ and an irreducible polynomial $f \in K[x]$. I have the following definition: a $K$-automorphism is a rational fraction $A \in K(x)$, such that for some rational fraction $D(x) \in K(x)$, we have $f(A(x)) = D(x)f(x)$.

Note that $K(x)$ is the field of rational functions in $x$. My questions are:

1) Is a $K$-automorphism a Galois action, i.e., an action of the Galois group of $f$ on $K$ ? or can there be some actions that are not of this form ?

2) Why are $A, D$ rational fractions ?

3) Could I defined the order of a Galois action?

Any links to litterature concerning the subject or details about this definition are more than welcome !

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    Can you define your terms? I assume $K$ is a field, what is $f$ supposed to be? Is $K(x)$ the field of fractions of the polynomial ring over $K$? – 2017-02-10
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    Sorry, I forgot! I will update right away – 2017-02-10
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    Something looks wrong here. Ordinarily, an automorphism is not an element of any field. Did you get this definition out of some source? – 2017-02-12

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