Symbolic solver complete with a high-level interactive language, user extensible library and visualisation tool for the manipulation of graphical output. For Unix and Windows. No longer produced: support ceases at the end of 2001.
http://www.nag.co.uk/symbolic_software.asp
Savannah Free Software project for revival of Axiom as open source.
http://savannah.nongnu.org/projects/axiom
A Computational Algebra system for algebra, number theory and geometry.
http://magma.maths.usyd.edu.au/
Computer algebra system for computation in commutative and non-commutative rings and modules. Win32, Unix.
http://felix.hgb-leipzig.de/
The FRISCO project aims to develop highly efficient, versatile polynomial solvers for industrial users.
http://www.nag.co.uk/projects/FRISCO.html
Package for Algebraic Function Fields in one variable. Based on Axiom. Source code is available.
http://www-rocq.inria.fr/codes/Gaetan.Hache/PAFF.html
A collection of routines, written in C, for computing in the representation theory of classical and symmetric groups, and related areas. Public domain.
http://www.mathe2.uni-bayreuth.de/axel/symneu_engl.html
Macaulay is a computer algebra system for mathematical computations in algebraic geometry and commutative algebra. At its core is a carefully tuned implementation of Grobner basis methods for manipulating systems of polynomial equations.
http://www.math.columbia.edu/~bayer/Macaulay.html