This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling.

The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author's lectures at Purdue University over the last few years.

Contents:

• Quadratic Equations (Rings)
• Curves and Surfaces (Fields)
• Tangents and Polars (Valuations)
• Varieties and Models (Ideals)
• Projective Varieties (Modules)
• Pause and Refresh (Groups)

“I cannot imagine that any mathematician would read even a few pages of this book without learning something new or seeing a new connection or gaining an insight into something that they thought they already knew. And between those insights and his writing, you will soon find yourself captivated by Abhyankar as well. I recommend that anyone with an interest in algebra find a copy of this book and flip through it.”

“What is the book good for? The book covers a lot of material for several courses like basic commutative algebra, commutative algebra or algebraic geometry, as well as additional material for basic courses in algebra. It might be used as an encyclopedia for commutative ring theory with a view towards algebraic geometry … it might be used as a source for inspiration for additional, interesting themes for courses as mentioned above … It might be recommended to anybody who is willing to see the fascinating, concrete as well as abstract development of algebra during the last centuries, including its applications to algebraic geometry and its potential … It is a consequence of the author’s masterpiece to bring together so many aspects of commutative ring theory. All together, it is not a classical textbook in the style ‘groups, rings, fields’. It is a unique, original exposition full of valuable insights.”

“The result is an incomparable new textbook of abstract algebra, complementing the existing literature in a highly interesting, alternative and utmost valuable manner. Also experienced teachers and researchers in the field can profit a great deal from the author’s unique textbook.”