Algebra

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).

• 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.

'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