Algebra
Notes from the Underground

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

Paolo Aluffi (Author)

9781108958233, Cambridge University Press

Paperback / softback, published 3 June 2021

488 pages
24.4 x 17.1 x 2.7 cm, 0.84 kg

'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

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.

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.

Subject Areas: Algebra [PBF]