A note on linear automorphisms over $\textbf {R}$

Abstract:

Let $K$ be a rational (= purely transcendental) extension of (the field) $k$, and let $s$ be a $k$-automorphism of $K$ of finite order. Let $s$ be linear in the sense that $K$ has a base $B$ (i.e., a transcendence basis $B$ with $K = k(B)$) for which the $k$-submodule $\Sigma (kb:b \in B)$ of $K$ generated by $B$ is stabilized by $s$. In [1, Question 6], it is asked whether $s$ is completely determined by its order (and $\operatorname {tr} .{\deg _k}(K)$) and it is proved that, when $k$ is the complex number field ${\mathbf {C}}$, then the answer to this question is affirmative iff $\operatorname {tr} .{\deg _{\mathbf {C}}}(K) > 1$ [1, Corollary 9, Question 6 and Lemma 7]. In this paper, we solve the problem for the field ${\mathbf {R}}$ of real numbers under the condition that $\operatorname {tr} .{\deg _{\mathbf {R}}}(K)$ is $\ne 2,3$. For $\operatorname {tr} .{\deg _{\mathbf {R}}}(K) = 2\;{\text {or}}\;3$, the problem remains open.