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.
https://en.wikipedia.org/wiki/Eisenstein%27s_criterion
https://en.wikipedia.org/wiki/Cohn%27s_irreducibility_criterion
See section Dumas's criterion in book Polynomials by Prasolov.
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.
See Theorem 1 in http://cms.dm.uba.ar/academico/materias/2docuat2011/teoria_de_numeros/Irreducible.pdf.
See Theorem 2.2.7 ([Os1]) in book Polynomials by Prasolov.
https://en.wikipedia.org/wiki/Perron%27s_irreducibility_criterion
See Theorem 2.2.8 (Polya) in book Polynomials by Prasolov.
See Theorem 2 in Irreducibility of Polynomials with Low Absolute Values by R. J. Levit.
See Schur's theorem for example in http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/schurtheorem.pdf.
See Theorem 2.2.6 ([Br]) in book Polynomials by Prasolov.
See https://arxiv.org/pdf/1304.0874.pdf advanced use of newton polygons.
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/).
Install required Python dependencies using:
python -m pip install -r requirements.txt
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.).