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Table of Contents

Part I. Rings: 1. The integers; 2. Modular arithmetic; 3. Rings; 4. The category of rings; 5. Canonical decomposition, quotients, and isomorphism theorems; 6. Integral domains; 7. Polynomial rings and factorization; Part II. Modules: 8. Modules and abelian groups; 9. Modules over integral domains; 10. Abelian groups; Part III. Groups: 11. Groups - preliminaries; 12. Basic results on finite groups; Part IV. Fields: 13. Field extensions; 14. Normal and separable extensions, and splitting fields; 15. Galois theory.

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A conversational introduction to abstract algebra from a modern, rings-first perspective, including a treatment of modules.

About the Author

Paolo Aluffi is Professor of Mathematics at Florida State University. Aluffi earned a Ph.D. from Brown University with a dissertation on the enumerative geometry of plane cubic curves, under the supervision of William Fulton. His research interests are in algebraic geometry, particularly intersection theory and its application to the theory of singularities and connections with theoretical physics. He has authored about 70 research publications and given lectures on his work in 15 countries. Beside Notes from the Underground, he has published a graduate-level textbook in algebra (Algebra: Chapter 0, AMS) and a mathematics book for the general public, in Italian (Fare matematica, Aracne Editrice).


'I highly recommend this excellent textbook for undergraduate algebra classes. The book features the 'rings-first' approach which I personally strongly prefer to the traditional 'groups-first' in an undergraduate class. The exposition is highly conceptual, mathematically neat and rigorous. A slight bias toward algebraic geometry and number theory is another advantage. The textbook can be fully covered within two semesters of undergraduate algebra providing the students with a rock solid foundation for further education in any algebra-related area of math: the breadth and depth of the coverage of all standard undergraduate algebra ideas and topics are just right.' Pavel Guerzhoy, University of Hawaii at Manoa
'Aluffi's Algebra: Notes from the Underground will modernize the teaching of undergraduate algebra, much like his earlier book Algebra: Chapter 0 did for graduate algebra. The book emphasizes the aspects and mindset of the subject most relevant to current researchers in mathematics. The standard material covered in undergraduate algebra is very well-presented and is supplemented with numerous interesting applications to other fields such as algebraic geometry and number theory. The book maintains a conversational tone throughout and teaches the reader universal properties and the modern categorical viewpoint by infusion. This is an excellent book for a year-long undergraduate course in algebra and will bridge the gap between undergraduate and graduate courses on the subject. A wealth of well-chosen exercises and detailed solutions make this book an invaluable resource for self-study.' Izzet Coskun, University of Illinois at Chicago
'Algebra: Notes from the Underground is the modern introduction to abstract algebra that mathematics has been missing. Paolo Aluffi invites students to fall in love with rings, modules, abelian groups, groups, and fields through a beautifully written tour that is equally suited for self-study or for a two-semester undergraduate course.' Emily Riehl, Johns Hopkins University

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