Description
Book SynopsisFrom 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 lo
Trade Review'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
Table of ContentsPart 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.
