nontrivial $K$-automorphism of $K(x)$

Question

How can I find $K$-automorphism $\sigma \in \mathrm{Aut}(K(x))$ different from identity such that $\sigma (x(x+1))=x(x+1)$?

Answer 1

HINT: The $K$-automorphisms of $K(x)$ are precisely the maps of the form $x\ \mapsto\ ax+b$ with $a\in K^{\times}$ and $b\in K$. Hence you want to find a solution $(a,b)\neq(1,0)$ to the equation $$(ax+b)(ax+b+1)=x(x+1).$$