Algebra: Notes from the Underground (Cambridge Mathematical Textbooks)
Cambridge University Press, 6/3/2021
EAN 9781108958233, ISBN10: 1108958230
Paperback, 488 pages, 24.4 x 17 x 2.8 cm
Language: English
Originally published in English
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.