Math 6310 • Algebra

Let \(k\) be a field and \(k(x)\) the field of rational functions over \(k\), in the indeterminate
\(x\). Consider the automorphisms \(\sigma\) and \(\tau\) of \(k(x)/k\) defined by
\[
\sigma(x)=\frac{1}{1-x}\qquad\text{and}\qquad \tau(x)=\frac{1}{x}.
\]
\begin{enumerate}
\item Prove that the subgroup \(G=\langle\sigma,\tau\rangle\) of \(\operatorname{Aut}(k(x)/k)\)
is isomorphic to \(S_3\).
\item Prove that the fixed subfield under the action of \(G\) equals \(k(t)\), where
\[
t=\frac{(x^2-x+1)^3}{x^2(x-1)^2}.
\]
\end{enumerate}