Paolo Aluffi

Algebra (eBook, PDF)

Notes from the Underground

• Format: PDF

Produktbeschreibung

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 naive set theory.

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Produktdetails

• Verlag: Cambridge University Press
• Erscheinungstermin: 3. Juni 2021
• Englisch
• ISBN-13: 9781108962698
• Artikelnr.: 66183957

Autorenporträt

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).

Inhaltsangabe

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.