Let $p$ be a prime number. We have a bijection $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ where the $a_i \in \{0, \ldots, p-1\}$. It maps pre-periodic expansions to pre-periodic expansions, so induces a bijection $\mathbb Q \to \mathbb F_p(t)$. It does not respect addition, multiplication, additive inverse, etc.

Is there any algebraic irrational $\alpha \in \mathbb Q_p/\mathbb Q$ such that $\sigma(\alpha)$ is an algebraic element of the field extension $\mathbb F_p(\!(t)\!)/\mathbb F_p(t)$?

I'm especially interested in the preimage of quadratic irrational $\beta$, because those have periodic continued fraction expansion. Also, because $\sigma$ is an isometry (with the RHS given the metric from the $t$-adic valuation), this gives good rational approximations for $p$-adic numbers.

I know little about telling from the series expansion whether something is algebraic. All I know is that algebraic elements (of degree $n$) of $\mathbb F_p(\!(t)\!)$ have finite irrationality measure (at most $n$), if we define it analogously as $$\sup \left\{ \mu>0 : \exists (p,q) \in \mathcal O^2 \text{ with }|q|\text{ arbitrarily large} : \left|\alpha - \frac pq\right| \leq |q|^{-\mu} \right\}$$ where $\mathcal O = \mathbb F_p[t^{-1}]$. (I don't know if this is the "correct" generalization). This means that elements like $\sum_{n \geq 0}t^{n^n}$ are transcendental .

Notes: (*) The continued fraction expansion of a Laurent series is defined analogously to the usual definition of continued fractions, from the notion of "integral part" $\sum_{i \leq 0} a_i t^i$ and it has very nice properties, especially over finite fields.

(*) There is something called Ruban's continued fraction of $p$-adic numbers, which is defined from the notion of integral part $\sum_{i \leq 0} a_i p^i$, and it doesn't have those nice properties.

(*) What I'm curious about is what happens when we consider the CF expansion of $\sigma(\alpha)$.


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