Software

Albert
Albert is an interactive program to assist the specialist in the study of nonassociative algebra: the main problem addressed is the recognition of polynomial identities in varieties of nonassociative algebras. Download source code in C.
http://www.cs.clemson.edu/~dpj/albertstuff/albert.html

Aldor
Originally an extension to Axiom, now a separate product. Binaries are available free for non-profit use, with documentation, tips, projects and links.
http://www.aldor.org/

Algebra Cheat
Program solves variable equations and gives explanations of steps used.
http://www.bacsoftware.co.uk/algebra/

Algebra Helper
Helps students with equations and inequalities, simplifying expression, graphing and complex numbers. Includes problem solving exercises.
http://www.algebra-answer.com/

Algebra Problem Solver
Personal Algebra Tutor - Enter your algebra problems and get instant step-by-step solutions with explanations.
http://www.cyberedinc.com/

Algebra Word Problem Tutor:Miss Lindquist
The world's first web-delivered self-improving intelligent tutoring system able to tutor students in writing expressions for algebra word problems.
http://www.cs.cmu.edu/~neil/

AXIOM
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

Axiom Computer Algebra System
Savannah Free Software project for revival of Axiom as open source.
http://savannah.nongnu.org/projects/axiom

CMAT
A matrix calculator program, written in C. Calculations can be performed on matrices with complex rational coefficients using exact arithmetic routines, as well as on matrices with elements mod p.
http://www.numbertheory.org/cmat/krm_cmat.html

CoCoA
A special-purpose system for computations in Commutative Algebra.
http://cocoa.dima.unige.it/

Combinatorial Representation Theory (CREP)
CREP is designed to deal with categories whose morphism spaces are finite-dimensional over a field k. The main example of a category with this property is the category of finite-dimensional representations of an associative unital k-algebra.
http://www.mathematik.uni-bielefeld.de/~sek/crep.html

Comparison of Polynomial-Oriented Computer Algebra Systems
By Robert H. Lewis and Michael Wester. Presented as a poster to the 1999 ISSAC Conference.
http://www.fordham.edu/lewis/cacomp.html

Computer Algebra and Problem Solving Environments
Abstract for Stanly Steinberg's article in "Comparative CAS Reviews and Philosophy", suggesting 6 ways in which computer algebra systems could be improved.
http://math.unm.edu/~wester/aca96/Steinberg_abstract.html

Computer Algebra Benchmarks
Collated by Computer Algebra Fachgruppe, Mannheim.
http://krum.rz.uni-mannheim.de/cafgbench.html

Computer Algebra Pages and Servers
A collection of links.
http://krum.rz.uni-mannheim.de/cabench/cawww.html

Felix
Computer algebra system for computation in commutative and non-commutative rings and modules. Win32, Unix.
http://felix.hgb-leipzig.de/

Fermat
Computer algebra system that does arithmetic of arbitrarily long integers and fractions, symbolic calculations, graphics, and other numerical calculations. Free download. Documentation.
http://www.bway.net/~lewis/

FINITAS
Discussion group for this package for the construction and analysis of finite algebraic structures such as loops, quasigroups, and groups.
http://groups.yahoo.com/group/finitas/

FRISCO - A Framework for Integrated Symbolic/Numeric Computation
The FRISCO project aims to develop highly efficient, versatile polynomial solvers for industrial users.
http://www.nag.co.uk/projects/FRISCO.html

Galois Field Arithmetic Library
A simple opensource library in C++ for arithmetic between elements and polynomials over finite fields.
http://www.partow.net/projects/galois/

GAP - Groups, Algorithms and Programming
A free system for computational discrete algebra.
http://www-history.mcs.st-and.ac.uk/~gap/

GB and FGB
Software for solving algebraic equations: GB computes Groebner bases and FGB solves polynomial systems.
http://www-calfor.lip6.fr/~jcf/

GIAC
Is A Computer algebra system. A semi-classical CAS by Bernard Parisse, available for Windows, Linux.
http://www-fourier.ujf-grenoble.fr/~parisse/english.html

GiNaC
An open framework for symbolic computation within the C++ programming language. The name GiNaC is an iterated and recursive abbreviation for "GiNaC is Not a CAS", where CAS stands for Computer Algebra System. Its design is revolutionary in a sense that contrary to other CAS it does not try to provide extensive algebraic capabilities and a simple programming language but instead accepts a given language (C++) and extends it by a set of algebraic capabilities. It is published under the GNU Public License (GPL).
http://www.ginac.de/

JACAL
An interactive symbolic mathematics program. JACAL can manipulate and simplify equations, scalars, vectors, and matrices of single and multiple valued algebraic expressions containing numbers, variables, radicals, and algebraic differential, and holonomic functions. Linux RPM distribution.
http://swissnet.ai.mit.edu/~jaffer/JACAL.html

KAN
Rings and Computer algebra software, part of OpenXM.
http://www.math.sci.kobe-u.ac.jp/KAN/

LAPACK -- Linear Algebra PACKage
written in Fortran77; provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems.
http://netlib2.cs.utk.edu/lapack/

LAPACK++: Linear Algebra Package in C++
(v. 1.1a) a software library for numerical linear algebra; however, it does not include all of the capabilities of original f77 LAPACK.
http://math.nist.gov/lapack++/

Macaulay
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/index.html

Macaulay 2
A new software system devoted to supporting research in algebraic geometry and commutative algebra.
http://www.math.uiuc.edu/Macaulay2/

MAS - The Modula-2 Algebra System
MAS is an experimental computer algebra system with algebraic specification capabilities for the design and study of algebraic algorithms. Downloads, documentation.
http://www.fmi.uni-passau.de/algebra/projects/mas.php3

Math Forum - Algebra Software
Algebra software resources compiled by The Math Forum.
http://mathforum.org/algebra/alg.software.html

Math Symbolic Library
Evaluate, calculate, simplify and derive symbolic scalar, complex or vector expression. [OpenSource]
http://www.cdrnet.net/projects/nmath/

MathPad - Scientific Matrices Calculator
Java applet based online Matrix calculation program for analyzing various simultaneous equations.
http://www.mathpad.com/public/htmls/main/left/matrix/MatrixInfo.html

Mathscribe
Mathscribe is free dynamic graphing and modeling software for use in algebra classes. Lesson plans for linears and quadratics are included.
http://www.mathscribe.com/

Matrix Expression Templates (MET)
C++ matrix class library which promotes the notational convenience of linear algebraic codes. Free download.
http://met.sourceforge.net/

Maxima
A GPL CAS based on DOE-MACSYMA. A GPL project.
http://maxima.sourceforge.net/

Online Gröbner Basis Calculations
Online computation of reduced ideal bases.
http://grobner.nuigalway.ie/grobner/basis.html

perisic
A Java package for multivariate polynomials over polymorphic rings.
http://ring.perisic.com/

Schur
A package for Lie groups by Brian G. Wybourne. DOS, Win32, Unix. Ordering information.
http://smc.vnet.net/Schur.html

SINGULAR
Computer Algebra System for polynomial computations with special emphasis on commutative algebra, algebraic geometry, and singularity theory.
http://www.singular.uni-kl.de/

Software for Computing Primitive Polynomials.
Free C software for computing primitive polynomials of degree n modulo p for p^n up to about 2^60.
http://www.seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/overview.html

SymbolicNet
Directory of links on Symbolic Algebraic Computation: Frequently Asked Questions, Live Demos, software for download, events, bibliography, research groups, and researcher directory.
http://www.symbolicnet.org/

Symmetrica
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

A Tutorial Introduction To AXIOM
By Martin N. Dunstan.
http://www.dcs.st-and.ac.uk/~mnd/documentation/axiom_tutorial/axiom.html

Yet Another Computer Algebra System
A general purpose easy program for It is built on top of its own programming language designed for symbolic manipulation of math symbols. Includes information on how to use the software.
http://yacas.sourceforge.net/

ZEN
A toolbox for computation in finite rings, by Florent Chabaud and Reynald Lercier. ANSI-C source, free download, GPL.
http://fchabaud.free.fr/English/ZEN/