For u ∉ Q̄ and coprime p, q ∈ Q̄[X]: if Q̄ + Q̄u + Q̄u⁻¹ + Σ Q̄ c (p(u)/q(u))ᵏ (k ≤ 3) is five-dimensional, then max(deg p, deg q) ≤ 2

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$, let $p, q \in \overline{\mathbb{Q}}[X]$ be coprime, $h = p(u)/q(u)$, let $c \in \mathbb{C}$ be non-zero, and let $W$ be the $\overline{\mathbb{Q}}$-span of $1, u, u^{-1}$ and $c, ch, ch^2, ch^3$. Suppose $\dim W = 5$. If $\max(\deg p, \deg q) \geq 1$, then $\max(\deg p, \deg q) \leq 2$.

So the ratio of a $4$-term progression in such a space generates a subfield of index $1$ or $2$ of \overline{\mathbb{Q}}$(u)$.

Proof. Evaluation at the transcendental $u$ embeds $\overline{\mathbb{Q}}(X)$ in $\mathbb{C}$; let $f = p/q$ there, so $f \mapsto h$. By coprimality, $\max(\deg p, \deg q) \le [\overline{\mathbb{Q}}(X) : \overline{\mathbb{Q}}(f)]$ (Mathlib's RatFunc.finrank_eq_max_natDegree, from the Lüroth file). $f$ is not constant, so $h$ is transcendental and $c, ch, ch^2, ch^3$ are free; with $\dim(\overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}u^{-1}) = 3$ and $\dim W = 5$, the intersection $H \cap P$ of the two parts has dimension at least $2$. Two non-proportional $x_i = cA_i(h)$ in it have $ux_i = R_i(u)$ with $\deg R_i \le 2$; $\varphi = R_1/R_2$ is non-constant and maps to $A_1(h)/A_2(h)$, so $\varphi \in \overline{\mathbb{Q}}(f)$ and $[\overline{\mathbb{Q}}(X) : \overline{\mathbb{Q}}(f)]$ divides $[\overline{\mathbb{Q}}(X) : \overline{\mathbb{Q}}(\varphi)] \le 2$.

Novelty. Not found in the sources read; it is short.