Singular

Singular is an open source computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory.

Singular provides:

highly efficient core algorithms, a multitude of advanced algorithms in the above fields, an intuitive, C-like programming language, easy ways to make it user-extendible through libraries, and a comprehensive online manual and help function.

Its main computational objects are ideals, modules and matrices over a large number of baserings. These include:

polynomial rings over various ground fields and some rings (including the integers), localizations of the above, a general class of non-commutative algebras (including the exterior algebra and the Weyl algebra), quotient rings of the above, tensor products of the above. Singular's core algorithms handle Gröbner resp. standard bases and free resolutions, polynomial factorization, resultants, characteristic sets, and numerical root finding.