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E-raamat: Algebra: Notes from the Underground

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(Florida State University)
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Algebra: Notes from the UndergroundSuurem pilt
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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.

Arvustused

'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 'Aluffi's Algebra: Notes from the Underground will modernize the teaching of undergraduate algebra, much like his earlier book Algebra: Chapter 0 did for graduate algebra. The book emphasizes the aspects and mindset of the subject most relevant to current researchers in mathematics. The standard material covered in undergraduate algebra is very well-presented and is supplemented with numerous interesting applications to other fields such as algebraic geometry and number theory. The book maintains a conversational tone throughout and teaches the reader universal properties and the modern categorical viewpoint by infusion. This is an excellent book for a year-long undergraduate course in algebra and will bridge the gap between undergraduate and graduate courses on the subject. A wealth of well-chosen exercises and detailed solutions make this book an invaluable resource for self-study.' Izzet Coskun, University of Illinois at Chicago 'Algebra: Notes from the Underground is the modern introduction to abstract algebra that mathematics has been missing. Paolo Aluffi invites students to fall in love with rings, modules, abelian groups, groups, and fields through a beautifully written tour that is equally suited for self-study or for a two-semester undergraduate course.' Emily Riehl, Johns Hopkins University 'For anyone planning to teach an undergraduate abstract algebra class anytime soon I strongly recommend the book under review. I have nothing but praise for this book. What really sets this text apart is Aluffi's writing style. His book has a remarkable narrative drive, and he is constantly reminding the reader of the big picture and tossing out tantalizing hints of what lies ahead.' John J. Watkins, MathSciNet

Muu info

A conversational introduction to abstract algebra from a modern, rings-first perspective, including a treatment of modules.
Introduction ix
Before we Begin xiii
Part I Rings
1(150)
1 The Integers
3(19)
1.1 The Well-Ordering Principle and Induction
3(3)
1.2 `Division with Remainder' in Z
6(1)
1.3 Greatest Common Divisors
7(5)
1.4 The Fundamental Theorem of Arithmetic
12(10)
Exercises
20(2)
2 Modular Arithmetic
22(19)
2.1 Equivalence Relations and Quotients
22(1)
2.2 Congruence mod n
23(3)
2.3 Algebra in Z/nZ
26(3)
2.4 Properties of the Operations +, · on Z/nZ
29(5)
2.5 Fermat's Little Theorem, and the RSA Encryption System
34(7)
Exercises
39(2)
3 Rings
41(18)
3.1 Definition and Examples
41(6)
3.2 Basic Properties
47(4)
3.3 Special Types of Rings
51(8)
Exercises
57(2)
4 The Category of Rings
59(18)
4.1 Cartesian Products
59(2)
4.2 Subrings
61(3)
4.3 Ring Homomorphisms
64(5)
4.4 Isomorphisms of Rings
69(8)
Exercises
75(2)
5 Canonical Decomposition, Quotients, and Isomorphism Theorems
77(29)
5.1 Rings: Canonical Decomposition, I
77(2)
5.2 Kernels and Ideals
79(4)
5.3 Quotient Rings
83(6)
5.4 Rings: Canonical Decomposition, II
89(2)
5.5 The First Isomorphism Theorem
91(2)
5.6 The Chinese Remainder Theorem
93(7)
5.7 The Third Isomorphism Theorem
100(6)
Exercises
103(3)
6 Integral Domains
106(23)
6.1 Prime and Maximal Ideals
106(4)
6.2 Primes and Irreducibles
110(2)
6.3 Euclidean Domains and PIDs
112(5)
6.4 PIDs and UFDs
117(3)
6.5 The Field of Fractions of an Integral Domain
120(9)
Exercises
126(3)
7 Polynomial Rings and Factorization
129(22)
7.1 Fermat's Last Theorem for Polynomials
129(2)
7.2 The Polynomial Ring with Coefficients in a Field
131(5)
7.3 Irreducibility in Polynomial Rings
136(3)
7.4 Irreducibility in Q[ x] and Z[ x]
139(4)
7.5 Irreducibility Tests in Z[ x]
143(8)
Exercises
148(3)
Part II Modules
151(76)
8 Modules and Abelian Groups
153(21)
8.1 Vector Spaces and Ideals, Revisited
153(5)
8.2 The Category of R-Modules
158(3)
8.3 Submodules, Direct Sums
161(3)
8.4 Canonical Decomposition and Quotients
164(4)
8.5 Isomorphism Theorems
168(6)
Exercises
171(3)
9 Modules over Integral Domains
174(28)
9.1 Free Modules
174(7)
9.2 Modules from Matrices
181(6)
9.3 Finitely Generated vs. Finitely Presented
187(3)
9.4 Vector Spaces are Free Modules
190(3)
9.5 Finitely Generated Modules over Euclidean Domains
193(4)
9.6 Linear Transformations and Modules over k[ t]
197(5)
Exercises
199(3)
10 Abelian Groups
202(25)
10.1 The Category of Abelian Groups
202(5)
10.2 Cyclic Groups, and Orders of Elements
207(5)
10.3 The Classification Theorem
212(5)
10.4 Fermat's Theorem on Sums of Squares
217(10)
Exercises
223(4)
Part III Groups
227(66)
11 Groups---Preliminaries
229(38)
11.1 Groups and their Category
229(6)
11.2 Why Groups? Actions of a Group
235(6)
11.3 Cyclic, Dihedral, Symmetric, Free Groups
241(12)
11.4 Canonical Decomposition, Normality, and Quotients
253(7)
11.5 Isomorphism Theorems
260(7)
Exercises
264(3)
12 Basic Results on Finite Groups
267(26)
12.1 The Index of a Subgroup, and Lagrange's Theorem
267(2)
12.2 Stabilizers and the Class Equation
269(5)
12.3 Classification and Simplicity
274(2)
12.4 Sylow's Theorems: Statements, Applications
276(4)
12.5 Sylow's Theorems: Proofs
280(2)
12.6 Simplicity of JKn
282(4)
12.7 Solvable Groups
286(7)
Exercises
289(4)
Part IV Fields
293(84)
13 Field Extensions
295(22)
13.1 Fields and Homomorphisms of Fields
295(4)
13.2 Finite Extensions and the Degree of an Extension
299(2)
13.3 Simple Extensions
301(6)
13.4 Algebraic Extensions
307(2)
13.5 Application: `Geometric Impossibilities'
309(8)
Exercises
314(3)
14 Normal and Separable Extensions, and Splitting Fields
317(23)
14.1 Simple Extensions, Again
317(3)
14.2 Splitting Fields
320(6)
14.3 Normal Extensions
326(2)
14.4 Separable Extensions; and Simple Extensions Once Again
328(5)
14.5 Application: Finite Fields
333(7)
Exercises
337(3)
15 Galois Theory
340(37)
15.1 Galois Groups and Galois Extensions
340(5)
15.2 Characterization of Galois Extensions
345(5)
15.3 The Fundamental Theorem of Galois Theory
350(7)
15.4 Galois Groups of Polynomials
357(4)
15.5 Solving Polynomial Equations by Radicals
361(9)
15.6 Other Applications
370(7)
Exercises
373(4)
Appendix A Background 377(25)
Appendix B Solutions to Selected Exercises 402(59)
Index of Definitions 461(2)
Index of Theorems 463(2)
Subject Index 465
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).
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