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Math-1502 · mathematics · Graph theory

O17 — Constructing irreducible polynomials over finite fields

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $p$ be a prime and $d\ge1$ an integer. A monic polynomial $F\in\mathbb F_p[X]$ of degree $d$ is irreducible if it cannot be written as a product of two positive-degree polynomials over $\mathbb F_p$. Such a polynomial presents the extension field with $p^d$ elements as $\mathbb F_p[X]/(F)$.

Can one construct an irreducible polynomial of degree exactly $d$ by a deterministic classical algorithm using a number of bit operations bounded by a fixed polynomial in $d$ and $\log p$? The output is the full list of its $d+1$ coefficients, represented by binary residues. Accordingly, the degree parameter is measured in unary, or equivalently the complexity bound is explicitly polynomial in the numerical degree $d$; polynomial time in $\log d$ would not even permit writing the required general dense output.

No polynomial to be factored is part of the input, and the algorithm may choose any irreducible polynomial of the requested degree. The same uniform construction must work as both the characteristic and degree vary. Neither randomness nor an oracle for selecting suitable elements of the field is allowed without accounting for its deterministic implementation.

The question concerns an entirely deterministic procedure. A randomized algorithm whose outputs usually agree with one canonical polynomial is still randomized and does not, on that property alone, supply the requested construction.

Short English statement

Can an irreducible polynomial of any prescribed degree over any prime field be constructed deterministically in time polynomial in the degree and logarithm of the prime?

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8400019 (AMR-083-0019), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Adleman & McCurley - Open Problems in Number Theory Complexity II (1994), O17, p14 / LNCS p304. Original source indexed by AMR-083-0019. https://mccurley.org/papers/open.ps.gz.
  • S3 S. S. Rai, Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields (2024). A randomized notion distinct from deterministic construction. https://arxiv.org/abs/2410.04071.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1502-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1502

Prime-field factorization permits the quadratic nonresidue construction and hence the degree-two irreducible case. It does not construct irreducible polynomials of every prescribed degree as requested here.

O17 — Constructing irreducible polynomials over finite fields

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 142; 2026-10-04), the asserted result is that dense polynomials over binary-specified prime fields have uniform deterministic complete factorization in bit time polynomial in degree and $\log p$, using the cited Hecke zero-free input.

Mathematical statement of the claimed result

Manuscript 1.
For any binary-encoded prime $p$ and nonzero densely encoded $f\in\mathbb F_p[x]$ of degree $n$, a uniform deterministic algorithm is claimed to output \[ f=c\prod_i g_i^{e_i} \] with $c$ the leading coefficient, distinct monic irreducible $g_i$ and positive integer multiplicities, in $O(((n+1)\lceil\log_2p\rceil)^{10^{12}})$ bit operations. Constants have an empty factor list. The computation is unconditional and deterministic; it receives neither an integer-factorization service nor a primitive-root oracle.

The separate algebraic reduction sets $L=\lceil\log_2p\rceil$, $B=20+(n+1)(L+1)$. If $p>B^{200000}$ and $n\ge2$, supply for every prime $q\le n$ a prime $\ell_q\notin\{2,3,p,q\}$ with \[ \ell_q\equiv1\pmod{12q},\qquad p^{(\ell_q-1)/q}\not\equiv1\pmod{\ell_q}. \] That reduction has bit cost $O((B+E)^C)$ for absolute $C$, where $E=\max(2,\max_q\ell_q)$, and $E=2$ when no such primes are required. This bound depends on the numerical size of the auxiliary primes; the full theorem additionally constructs sufficiently small primes.

Outline of the argument

For manuscript 1 (family 142; 2026-10-04), primary generators and norm equations reduce splitting to an odd-degree separator. Lifted ramification classes are filtered to keep extension degrees polynomial, and divisors at infinity distinguish roots; the analytic input bounds the auxiliary primes required by the finite algebraic algorithm.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 142; 2026-10-04).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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