Hermite's problem

Call a sequence $(a_n)_{n\ge 0}$ of natural numbers eventually periodic if there exist $N\ge 0$ and $p\ge 1$ such that $a_{n+p}=a_n$ for every $n\ge N$. Call a real number $x$ algebraic of degree $d$ if $x$ is a root of an irreducible polynomial of degree $d$ with rational coefficients; thus $x$ is algebraic of degree $1$ exactly when $x\in\mathbb{Q}$, algebraic of degree $2$ exactly when $x$ is a quadratic irrational, and algebraic of degree $3$ exactly when $x$ is a cubic irrational.

There exists a map

$$\Phi:\mathbb{R}\longrightarrow \mathbb{N}^{\mathbb{Z}_{\ge 0}},\qquad \Phi(x)=(a_n(x))_{n\ge 0},$$

computed by a single algorithm whose only input is the real number $x$, such that:

1. $\Phi$ is injective, i.e. each sequence in the image of $\Phi$ determines a unique real number; 2. for every $x\in\mathbb{R}$, the sequence $\Phi(x)$ is eventually periodic if and only if $x$ is algebraic of degree $3$.

More generally, for every integer $d\ge 1$ there exists a map $\Phi_d:\mathbb{R}\to\mathbb{N}^{\mathbb{Z}_{\ge 0}}$, computed by a single algorithm whose only input is $x$, such that $\Phi_d$ is injective and, for every $x\in\mathbb{R}$, the sequence $\Phi_d(x)$ is eventually periodic if and only if $x$ is algebraic of degree $d$.

(For $d=1$ the decimal expansion $x=a_0.a_1a_2a_3\ldots$, with $a_0\in\mathbb{Z}$ the integer part of $x$ and $a_1,a_2,\ldots\in\{0,1,\dots,9\}$, has this property, since $x=\sum_{n\ge 0}a_n/10^{n}$ is rational precisely when its digit sequence is eventually periodic; for $d=2$ the simple continued fraction expansion has this property.)

A related two-dimensional formulation: for the multidimensional continued fraction algorithm that assigns to each pair of real numbers $(x,y)$ a sequence of natural numbers, this sequence is eventually periodic if and only if $x$ and $y$ lie in one and the same cubic number field.