Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Communtative Algebra by David Cox, John Little, and Donal OShea.
From the introduction:
We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra. Until recently, these topics involved a lot of abstract mathematics and were only taught in graduate school. But in the 1960s, Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms, the last two decades have seen a minor revolution in commutative algebra. The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand, and has changed the practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these techniques in a whole range of problems.
The authors do presume students “…have access to a computer algebra system.”
The Wikipedia List of computer algebra systems has links to numerous such systems. A large number of which are free.
That list is headed by Axiom (Wikipedia article) and is an example of literate programming. The Axiom documentation looks like a seriously entertaining time sink! You may want to visit http://axiom-developer.org/
I haven’t installed Axiom so take that as a comment on its documentation more than its actual use. Use whatever system you like best and fits your finances.
I first saw this in a tweet from onepaperperday.
Enjoy!