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ArticlePublished 7 Aug 20262 min readBy Kevin Jogindistinct degree factorisationFrobeniusfinite fieldirreducibility test
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Polynomial Factorisation

Distinct Degree Factorisation

Separating irreducible factors by degree using GCDs against Frobenius powers, and the early-abort strategies that make it fast.

Engineering / MathematicsPolynomial Factorisation2 min readKV-MATH-0564

Distinct degree factorisation separates the irreducible factors of a squarefree polynomial according to their degree. It is deterministic and rests on a single clean fact about finite fields.

The governing fact

The polynomial X to the q to the d, minus X, is exactly the product of all monic irreducible polynomials whose degree divides d.

X^(q^d) - X = product of all monic irreducibles of degree dividing dq is the field size.

Key point

Taking the GCD of the input with this polynomial, for d = 1, 2, 3, ... in turn, peels off the factors of each degree in ascending order. Because lower degrees have already been removed, the GCD at step d captures exactly the factors of degree d.

The algorithm

Distinct degree factorisation

  1. InitialiseStart with the squarefree input and the polynomial X.
  2. Apply FrobeniusRaise the current tracker to the q-th power modulo the remaining input.
  3. Take the GCDThe GCD with the remaining input is the product of all irreducible factors of the current degree.
  4. Divide outRemove that product from the remaining input.
  5. AdvanceIncrement the degree and repeat.

Pitfall

The Frobenius power must be computed modulo the remaining polynomial by repeated squaring, exactly as in root finding. The exponent q^d is astronomically large and the power can never be formed explicitly.

Early termination

Key point

Once the degree exceeds half the remaining degree, whatever remains must be irreducible — there is no room for two factors of that size. This halves the work in the common case and is the single most effective optimisation.

Irreducibility testing

The same machinery gives a fast irreducibility test without producing any factors: a polynomial of degree d is irreducible exactly when it divides the Frobenius polynomial at d and shares no factor with any smaller one, which reduces to checking the prime divisors of d.

Costs of the Frobenius-based tests
TaskCost
Full distinct degree factorisationUp to half the degree many Frobenius applications
Irreducibility test onlyOne Frobenius power plus a GCD for each prime divisor of the degree
Finding one rootA single GCD at degree one — see root finding

Precomputing Frobenius

Cost

Frobenius is a linear map on the quotient ring. Precomputing its matrix converts each application from a repeated squaring to a matrix-vector product, which pays off strongly when many applications are needed — see finite field arithmetic in practice.

Output

The output is a set of polynomials, each a product of irreducibles of a single known degree. Separating those requires the probabilistic stage — see equal degree splitting.

Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 3.4.3. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.

Related pages

  • Finding Primitive Roots and Generators
  • Squarefree Factorisation of Polynomials
  • Cantor-Zassenhaus Equal Degree Splitting

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