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ArticlePublished 7 Aug 20262 min readBy Kevin Joginnumber fieldpolynomial factorisationnormTrager
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Polynomial Factorisation

Factoring Polynomials over Algebraic Number Fields

Factoring polynomials whose coefficients lie in a number field, by reduction to the rational case via norms.

Engineering / MathematicsPolynomial Factorisation2 min readKV-MATH-0570

Factoring a polynomial over a number field is required for constructing relative extensions, splitting fields, and subfield computation. The standard method reduces it to factoring over the rationals by taking a norm.

The norm construction

Given a polynomial over a number field, take the product of its conjugates under all embeddings. The result has rational coefficients and its factorisation constrains that of the original.

N(f)(X) = product over embeddings s of s(f)(X)Computable as a resultant, without computing embeddings.

Key point

The norm is computed as a resultant of the polynomial with the field's defining polynomial, eliminating the field generator. No numerical embeddings are needed — see resultants.

The algorithm

Factoring over a number field by norms

  1. Ensure squarefreeOver the number field.
  2. Shift if necessarySubstitute X plus a multiple of the field generator until the norm is squarefree.
  3. Compute the normAs a resultant; a polynomial over the rationals.
  4. Factor the normUsing integer factorisation.
  5. Recover factorsEach rational factor of the norm gives a factor of the original by a GCD over the number field.
  6. Undo the shiftSubstitute back.

The shift

Pitfall

The method requires the norm to be squarefree, which is not automatic. A shift by a multiple of the field generator fixes it, and only finitely many shift values fail — but the shift increases coefficient sizes substantially, so a small shift should be sought rather than a random one.

Cost

Cost

The norm has degree equal to the product of the polynomial degree and the field degree, with correspondingly large coefficients. This is the dominant cost and it grows quickly, which is why the method is comfortable for small fields and painful for large ones.

Alternatives

Direct p-adic methods

Factor modulo a prime ideal and lift within the number field. Avoids the degree blow-up of the norm but requires prime decomposition machinery.

Lattice methods

Extend the LLL-based approach to number field coefficients, using the field's own lattice structure.

Applications

Factoring over a number field is how relative extensions are constructed, how subfields are identified, and how splitting fields are built for Galois group computation.

Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 3.5.6. 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

  • Number Fields: Definition and Basic Properties
  • The Subfield Problem
  • Factoring Polynomials over the Integers
  • Root Finding over the Reals and Complex Numbers

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