thesil / irreducibility_criteria_checker

Simple Python script for quick checking of irreducibility of polynomial with integer coefficients by testing various criteria.

Irreducibility Criteria Checker

This repository contains Python scripts that can be used to check various Polynomial irreducibility criteria. Main purpose is to find, understand and possibly automate as many criteria as possible.

Eisenstein irreducibility criterion

https://en.wikipedia.org/wiki/Eisenstein%27s_criterion

Cohn's irreducibility criterion

https://en.wikipedia.org/wiki/Cohn%27s_irreducibility_criterion

Dumas' criterion for Newton polygons

See section Dumas's criterion in book Polynomials by Prasolov.

Checking irreducibility in finite field Fp

Algorithmic approach is used to decompose the polynomial into irreducible factors in finite fields Fp for small primes p. If the polynomial is irreducible in any of them, or if degrees of irreducible factors are not compatible, irreducibility over integers is implied.

Murty's irreducibility criterion

See Theorem 1 in http://cms.dm.uba.ar/academico/materias/2docuat2011/teoria_de_numeros/Irreducible.pdf.

Osada's irreducibility criterion

See Theorem 2.2.7 ([Os1]) in book Polynomials by Prasolov.

Perron's irreducibility criterion

https://en.wikipedia.org/wiki/Perron%27s_irreducibility_criterion

Polya's irreducibility criterion

See Theorem 2.2.8 (Polya) in book Polynomials by Prasolov.

Levit's irreducibility criterion

See Theorem 2 in Irreducibility of Polynomials with Low Absolute Values by R. J. Levit.

Schur's irreducibility criterion

See Schur's theorem for example in http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/schurtheorem.pdf.

Brauer's irreducibility criterion

See Theorem 2.2.6 ([Br]) in book Polynomials by Prasolov.

Bonciocat's irreducibility criterion

See https://arxiv.org/pdf/1304.0874.pdf advanced use of newton polygons.

Filaseta's criteria

See https://doi.org/10.1016/j.jnt.2013.11.001 and https://oeis.org/A253280 for generalization of Cohn's irreducibility criterion. Values beyond base 10 are taken from works of Cole (https://scholarcommons.sc.edu/etd/1590/) and Dunn (https://scholarcommons.sc.edu/etd/2809/).

Dependencies

Install required Python dependencies using:

python -m pip install -r requirements.txt

Basic usage

Run test suite:

python -m unittest discover"

To run specific criterion check, try for example:

python criteria/eisenstein.py "x^3-2*x^2+2*x+2"

To run all currently supported criteria use:

python check_all.py "P(x)"

where P(x) is some univariate polynomial from Z[x], e.g.

python check_all.py "x^3-24*x^2-240*x-728"

This will also performs various substituions (e.g. shift f(x+c) or reciprocal polynomials.).