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This link explains $\operatorname{Aut}_{K}{K(x)}$.

And I want to know how to solve two problems below in the Hungerford's Algebra, p.256.

$7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ consisting of the three automorphisms induced by $x\rightarrow x$, $x\rightarrow 1_K / (1_K -x)$, $x\rightarrow (x-1_K)/x$. Then $G$ is a subgroup of $\operatorname{Aut}_{K}{K(x)}$. Determine the fixed field of $G$.

$8.$ Assume char $K=0$ and let $G$ be the subgroup of $\operatorname{Aut}_{K}{K(x)}$ that is generated by the automorphism induced by $x\rightarrow x+ 1_K$. Then $G$ is an infinite cyclic group. Determine the fixed field $E$ of $G$. What is $[K(x):E]$?

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For $7,$ $G$ is cyclic because for $f(x)=1_K / (1_K -x)$, $f^2(x)=(x-1_K)/x$ and $f^3(x)=x$ hold. So, the fixed field of $G$ is $$\big\{ \varphi(x)\in K(x) \:|\: \varphi(x)=\varphi(1_K / (1_K -x))\big\}.$$ If this is right, is there any good way to find the above field?

For $8$, $$E=\big\{\varphi(x)\in K(x) \: | \: \varphi(x)=\varphi(x+1_K) \big\},$$ If this is right, is there any good way to find the above field?

user26857
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1 Answers1

3

Hints.

  1. Try $K(x+f(x)+f^2(x))$.

  2. If $f(x)/g(x)=f(x+n)/g(x+n)$ for every $n\ge 1$, where $f,g\in K[x]$, then show that $f(x)/g(x)$ is in $K$.

user26857
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  • is also solved here: http://math.stackexchange.com/a/808476/121097, but I think my hint (which actually gives a more suggestive form of the generator) should be enough. 8. can be found here: http://math.stackexchange.com/questions/528034/fixed-field-of-automorphism?rq=1.
    • – user26857 Oct 25 '14 at 08:27