Deterministic polynomial factorization over prime fields

What OpenAI claims

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in (n+1) log p. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

Paper

1. Deterministic Polynomial Factorization over Prime Fields

   4 October 2026 · Preprint, not peer reviewed · PDF

   We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial f ∈ F_p[x] given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in (deg f+1) log p. The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.