Hermite's problem

Definitions and mathematical statement

For a positive integer $n$, let \[ A_n=\{x\in{\mathbb{R}}:x\text{ is algebraic over }{\mathbb{Q}},\ [{\mathbb{Q}}(x):{\mathbb{Q}}]=n\}. \] A sequence $(a_j)_{j\geq0}$ is eventually periodic if there exist integers $N\geq0$ and $d\geq1$ such that $a_{j+d}=a_j$ for every $j\geq N$. The literal source asks, for every $n$, for an assignment \[ F_n:{\mathbb{R}}\longrightarrow{\mathbb{N}}^{{\mathbb{N}}} \] whose output is eventually periodic exactly for $x\in A_n$. Degree is exactly $n$, not degree at most $n$, and the assignment may depend on $n$.

As written, the source imposes no algorithmic, dynamical, structural or naturality constraint on $F_n$. An arbitrary assignment may branch on membership in $A_n$, making this bare existence criterion fail to express the substantive Hermite problem. The supplied summary instead points to a natural continued-fraction-type expansion or multidimensional algorithm; its input/output rules and success criteria are not specified.

The degree-two benchmark is Lagrange's theorem: a nonterminating regular continued fraction is eventually periodic exactly for a real quadratic irrational. Rational numbers terminate under the usual convention. Extending every finite expansion by an eventually periodic tail would spoil an exact-degree-two criterion on all real numbers, so finite outputs or a rational-input encoding must be specified. The source's natural-number sequence formulation also needs any sign and initial-digit conventions. Thus that benchmark cannot silently be treated as a literal definition of $F_2$ on every real input.

The requested higher-degree result should provide the intended natural expansion with the exact periodicity characterization, rather than an unrestricted set-theoretic coding. No such algorithm or definition of natural is supplied.

Short English statement

Find a suitably natural expansion whose eventual periodicity characterizes real algebraic numbers of each exact degree. Without constraints on the assignment, the literal coding question is not the substantive Hermite problem.

Sources

• Local UnsolvedMath registry, permanent ID 9400051, exact source code catalogue entry, "Hermite's problem"; no proposer or year supplied. Registry status partially solved, classification PARTIAL-PROGRESS.
• Wikipedia contributors, List of unsolved problems in mathematics, Algebraic number theory bullet 51, wikitext line 64, pinned revision 1366636767, https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory. Supplied MediaWiki extraction accessed July 29, 2026. The entry is staged pending semantic deduplication and primary-source review.
• The supplied summary invokes Lagrange's continued-fraction theorem for degree two and describes the higher-degree natural multidimensional-expansion question as open, referring generally to Wikipedia's "Hermite's problem" without a primary citation or exact algorithmic formulation.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW. Neither staging nor the registry published flag establishes source/status review.