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Algebra Notes from the Underground (Paolo Aluffi) (.pdf
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Algebra Notes from the Underground (Paolo Aluffi) (.pdf
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Algebra
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, appendix with complete solutions to around 150 of the main text
problems, and appendix with general background on basic logic and naïve set theory.
Paolo Aluf 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).
CAMBRIDGE MATHEMATICAL TEXTBOOKS
Cambridge Mathematical Textbooks is a program of undergraduate and beginning graduate-
level textbooks for core courses, new courses, and interdisciplinary courses in pure and applied
mathematics. These texts provide motivation with plenty of exercises of varying difficulty, inter-
esting examples, modern applications, and unique approaches to the material.
Advisory Board
John B. Conway, George Washington University
Gregory F. Lawler, University of Chicago
John M. Lee, University of Washington
John Meier, Lafayette College
Lawrence C. Washington, University of Maryland, College Park
A complete list of books in the series can be found at www.cambridge.org/mathematics
Recent titles include the following:
Chance, Strategy, and Choice: An Introduction to the Mathematics of Games and Elections,
S. B. Smith
Set Theory: A First Course, D. W. Cunningham
Chaotic Dynamics: Fractals, Tilings, and Substitutions, G. R. Goodson
A Second Course in Linear Algebra, S. R. Garcia & R. A. Horn
Introduction to Experimental Mathematics, S. Eilers & R. Johansen
Exploring Mathematics: An Engaging Introduction to Proof,J.Meier&D.Smith
A First Course in Analysis,J.B.Conway
Introduction to Probability, D. F. Anderson, T. Seppäläinen & B . Valkó
Linear Algebra, E. S. Meckes & M. W. Meckes
A Short Course in Differential Topology, B. I. Dundas
Abstract Algebra with Applications, A. Terras
Algebra
PAOLO ALUFFI
Florida State University
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