Deterministic Polynomial Factorization over Prime Fields

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