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

'I absolutely loved Aluffi's treatment of algebra and category theory in Algebra: Chapter 0 as a graduate student, and was delighted to find the same themes shine through in Algebra: Notes from the Underground' Blake Farman, Louisiana Tech University

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]