This text covers topics in algebraic geometry and commutative algebra with a strong perspective
toward practical and computational aspects. The first four chapters form the core of the book.
A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material
once these chapters are covered. In addition to the fundamentals of algebraic geometry-the
elimination theorem the extension theorem the closure theorem and the Nullstellensatz-this
new edition incorporates several substantial changes all of which are listed in the Preface.
The largest revision incorporates a new Chapter (ten) which presents some of the essentials of
progress made over the last decades in computing Gröbner bases. The book also includes current
computer algebra material in Appendix C and updated independent projects (Appendix D). The book
may serve as a first or second course in undergraduate abstract algebra and with some
supplementation perhaps for beginning graduate level courses in algebraic geometry or
computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented
course. It is assumed that the reader has access to a computer algebra system. Appendix C
describes features of Maple(TM) Mathematica® and Sage as well as other systems that are most
relevant to the text. Pseudocode is used in the text Appendix B carefully describes the
pseudocode used. Readers who are teaching from Ideals Varieties and Algorithms or are
studying the book on their own may obtain a copy of the solutions manual by sending an email
to jlittle@holycross.edu. From the reviews of previous editions: ...The book gives an
introduction to Buchberger's algorithm with applications to syzygies Hilbert polynomials
primary decompositions. There is an introduction to classical algebraic geometry with
applications to the ideal membership problem solving polynomial equations and elimination
theory. ...The book is well-written. ...The reviewer is sure that it will be an excellent guide
to introduce further undergraduates in the algorithmic aspect of commutative algebra and
algebraic geometry. -Peter Schenzel zbMATH 2007 I consider the book to be wonderful. ... The
exposition is very clear there are many helpful pictures and there are a great many
instructive exercises some quite challenging ... offers the heart and soul of modern
commutative and algebraic geometry. -The American Mathematical Monthly
