From rings to modules to groups to fields, this undergraduate introduction to abstract algebra follows an unconventional path. The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics - such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory - are all treated in detail. Students will appreciate the text's conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic logic and naïve set theory.
- Treats all standard topics for a first course in detail, moving from rings to modules to groups to fields, which allows for a more gradual entry into the subject
- Emphasizes a modern perspective, with hints toward category theory, exposing students to advanced topics and preparing them for graduate courses and applications of algebra in other subjects
- Includes 400+ exercises, with 150 worked-out problems from the main text, and plenty of examples to help students better understand the material
- Uses a conversational style and strong narrative flow that makes the material easier for students to follow and appreciate as a whole
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Reviews & endorsements
'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
'For anyone planning to teach an undergraduate abstract algebra class anytime soon I strongly recommend the book under review. I have nothing but praise for this book. … What really sets this text apart is Aluffi's writing style. His book has a remarkable narrative drive, and he is constantly reminding the reader of the big picture and tossing out tantalizing hints of what lies ahead.' John J. Watkins, MathSciNet
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Product details
- Date Published: July 2021
- format: Paperback
- isbn: 9781108958233
- length: 488 pages
- dimensions: 244 x 171 x 27 mm
- weight: 0.84kg
- availability: Available
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.
Look Inside
Author
Paolo Aluffi, Florida State University
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).