Does $\mathbb{R}(X)$ have an automorphism of order 7?

Question

The question comes from a 2020 home exam of an undergraduate course in field theory and Galois theory and is as follows:

Let $\mathbb{R}(X)$ be the field of rational functions over $\mathbb{R}$.

Prove or disprove:

There is an automorphism $\sigma \in Aut(\mathbb{R}(X))$ such that the order of $\sigma$ is 7.

Some of the ideas, while trying to attack this question, were with this question. But, after discovering that an automorphism of a field extension of $\mathbb{R}$ doesn't necessarily preserve it (although an automorphism of $\mathbb{R}$ is the identity), I'm not sure if this is the way to go.

After more research online, I was surprised to find little to no information about similar questions.

I will greatly appreciate any information relevant to this question. Also, an interesting follow-up question:

• What is the automorphism group, $Aut(\mathbb{R}(X))$? What are the orders of automorphisms $\sigma \in Aut(\mathbb{R}(X))$?

Answer 1

For an arbitrary field $K$, the field automorphisms of $K(x)$ that fix $K$ pointwise are the linear fractional substitutions: $f(x) \mapsto f((ax+b)/(cx+d))$ for all $f \in K(x)$, where $a, b, c, d \in K$ and $ad - bc \not= 0$. This is just a substitution map replacing $x$ with $(ax+b)/(cx+d)$ in all rational functions. Such substitutions are determined by the matrix $(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})$ up to nonzero scaling, so the automorphisms of $K(x)$ fixing the elements of $K$ are isomorphic to the group ${\rm PGL}_2(K) = {\rm GL}_2(K)/K^\times$. We express this as: ${\rm Aut}(K(x)/K) \cong {\rm PGL}_2(K)$. This is related to Lüroth's Theorem, which says every field between $K$ and $K(x)$ is $K(f(x))$ for some rational function $f(x)$ in $K(x)$.

For some $K$ the field automorphisms of $K(x)$ don't all fix the elements of $K$, but when $K = \mathbf R$ it is true: every field automophism of $\mathbf R(x)$ fixes the elements of $\mathbf R$. See Theorem 5.1 here. So ${\rm Aut}(\mathbf R(x)) = {\rm Aut}(\mathbf R(x)/\mathbf R)$. Therefore your question amounts to asking when an element of ${\rm PGL}_2(\mathbf R)$ has order $7$.

By the way, it is not important to know first whether or not all field automorphisms of $\mathbf R(x)$ fix the constants because you could at least start by focusing on such automorphisms and see if you can find one like that with order $7$. If you find one, great! Only if you suspect there isn't one would it become important to know that automorphisms of $\mathbf R(x)$ fix the elements of $\mathbf R$. But also, to be honest, I suspect nearly anyone who asks about field automorphisms of a rational function field only cares about those that actually fix the constants, and the failure to specify in the problem that "automorphism" means "automorphism fixing $\mathbf R$" might have just been an oversight.