代数(英文版.第2版)

.2.10 the correspondence theorem
2.11 product groups
2.12 quotient groups
exercises
3 vector spaces
3.1 subspaces of rn
3.2 fields
3.3 vector spaces
3.4 bases and dimension
3.5 computing with bases
3.6 direct sums
3.7 infinite-dimensional spaces
exercises
4 linear operators
4.1 the dimension formula
4.2 the matrix of a linear transformation
4.3 linear operators
4.4 eigenvectors
4.5 the characteristic polynomial
4.6 triangular and diagonal forms
4.7 jordan form
exercises
5 applications of linear operators
5.1 orthogonal matrices and rotations
5.2 using continuity
5.3 systems of differential equations
5.4 the matrix exponential
exercises
6 symmetry
6.1 symmetry of plane figures
6.2 isometries
6.3 isometries of the plane
6.4 finite groups of orthogonal operators on the plane
6.5 discrete groups of isometries
6.6 plane crystallographic groups
6.7 abstract symmetry: group operations
6.8 the operation on cosets
6.9 the counting formula
6.10 operations on subsets
6.11 permutation representations
6.12 finite subgroups of the rotation group
exercises
7 more group theory
7.1 cayley's theorem
7.2 the class equation
7.3 p-groups
7.4 the class equation of the icosahedral group
7.5 conjugation in the symmetric group
7.6 normalizers
7.7 the sylow theorems
7.8 groups of order 12
7.9 the free group
7.10 generators and relations
7.11 the todd-coxeter algorithm
exercises
8 bilinear forms
8.1 bilinear forms
8.2 symmetric forms
8.3 hermitian forms
8.4 orthogonality
8.5 euclidean spaces and hermitian spaces
8.6 the spectral theorem
8.7 conics and quadrics
8.8 skew-symmetric forms
8.9 summary
exercises
9 linear groups
9.1 the classical groups
9.2 interlude: spheres
9.3 the special unitary group su2
9.4 the rotation group s03
9.5 one-parameter groups
9.6 the lie algebra
9.7 translation in a group
9.8 normal subgroups of sl2
exercises
10 group representations
10.1 definitions
10.2 irreducible representations
10.3 unitary representations
10.4 characters
10.5 one-dimensional characters
10.6 the regular representation
10.7 schur's lemma
10.8 proof of the orthogonality relations
10.9 representations of su2
exercises
11 rings
11.1 definition of a ring
11.2 polynomial rings
11.3 homomorphisms and ideals
11.4 quotient rings
11.5 adjoining elements
11.6 product rings
11.7 fractions
11.8 maximal ideals
11.9 algebraic geometry
exercises
12 factoring
12.1 factoring integers
12.2 unique factorization domains
12.3 gauss's lemma
12.4 factoring integer polynomials
12.5 gauss primes
exercises
13 quadratic number fields
13.1 algebraic integers
13.2 factoring algebraic integers
13.3 ideals in z
13.4 ideal multiplication
13.5 factoring ideals
13.6 prime ideals and prime integers
13.7 ideal classes
13.8 computing the class group
13.9 real quadratic fields
13.10 about lattices
exercises
14 linear algebra in a ring
14.1 modules
14.2 free modules
14.3 identities
14.4 diagonalizing integer matrices
14.5 generators and relations
14.6 noetherian rings
14.7 structure of abelian groups
14.8 application to linear operators
14.9 polynomial rings in several variables
exercises
15 fields
15.1 examples of fields
15.2 algebraic and transcendental elements
15.3 the degree of a field extension
15.4 finding the irreducible polynomial
15.5 ruler and compass constructions
15.6 adjoining roots
15.7 finite fields
15.8 primitive elements
15.9 function fields
15.10 the fundamental theorem of algebra
exercises
16 galois theory
16.1 symmetric functions
16.2 the discriminant
16.3 splitting fields
16.4 isomorphisms of field extensions
16.5 fixed fields
16.6 galois extensions
16.7 the main theorem
16.8 cubic equations
16.9 quartic equations
16.10 roots of unity
16.11 kummer extensions
16.12 quintic equations
exercises
appendix
background material
a.1 about proofs
a.2 the integers
a.3 zorn's lemma
a.4 the implicit function theorem exercises
bibliography
notation
index

.　　1. Treat the abstract material with a light touch. You can have another go at it in Chapters 6 and 7.
　　2. For examples, concentrate on matrix groups. Examples from symmetry are best deferred to Chapter 6.
　　3. Don't spend much time on arithmetic; its natural place in this book is in Chapters 12 and 13.
　　4. De-emphasize the quotient group construction.
　　Quotient groups present a pedagogical problem. While their construction is conceptually difficult, the quotient is readily presented as the image of a homomorphism in most elementary examples, and then it does not require an abstract definition. Modular arithmetic is about the only convincing example for which this is not the case. And since the integers modulo n form a ring, modular arithmetic isn't the ideal motivating example for quotients of groups. The first serious use of quotient groups comes when generators and relations are discussed in Chapter 7. I deferred the treatment of quotients to that point in early drafts of the book, but, fearing the outrage of the algebra community, I eventually moved it to Chapter 2. If you don't plan to discuss generators and relations for groups in your course, then you can defer an in-depth treatment of quotients to Chapter 11, Rings, where they play a central role, and where modular arithmetic becomes a prime motivating example.
　　In Chapter 3, Vector Spaces, I've tried to set up the computations with bases in such a way that the students won't have trouble keeping the indices straight. Since the notation is used throughout the book, it may be advisable to adopt it.
　　The matrix exponential that is defined in Chapter 5 is used in the description of oneparameter groups in Chapter 10, so if you plan to include one-parameter groups, you will need to discuss the matrix exponential at some point. But you must resist the temptation to give differential equations their due. You will be forgiven because you are teaching algebra. Except for its first two sections, Chapter 7, again on groups, contains optional material. A section on the Todd-Coxeter algorithm is included to justify the discussion of generators and relations, which is pretty useless without it. It is fun, too.
　　There is nothing unusual in Chapter 8, on bilinear forms. I haven't overcome the main pedagogical problem with this topic - that there are too many variations on the same theme, but have tried to keep the discussion short by concentrating on the real and complex cases. In the chapter on linear groups, Chapter 9, plan to spend time on the geometry of SU2.
　　My students complained about that chapter every year until I expanded the section on SU2, after which they began asking for supplementary reading, wanting to learn more. Many of our students aren't familiar with the concepts from topology when they take the course, but I've found that the problems caused by the students' lack of familiarity can be managed.
　　Indeed, this is a good place for them to get an idea of a manifold.
　　I resisted including group representations, Chapter 10, for a number of years, on the grounds that it is too hard. But students often requested it, and I kept asking myself: If the chemists can teach it, why can't we? Eventually the internal logic of the book won out and group representations went in. As a dividend, hermitian forms got an application.
　　You may find the discussion of quadratic number fields in Chapter 13 too long for a general algebra course. With this possibility in mind, I've arranged the material so that the end of Section 13.4, on ideal factorization, is a natural stopping point.
　　It seemed to me that one should mention the most important examples of fields in a beginning algebra course, so I put a discussion of function fields into Chapter 15. There is always the question of whether or not Galois theory should be presented in an undergraduate course, but as a culmination of the discussion of symmetry, it belongs here. Some of the harder exercises are marked with an asterisk.
　　Though I've taught algebra for years, various aspects of this book remain experimental, and I would be very grateful for critical comments and suggestions from the people who use it.
　　ACKNOWLEDGMENTS
　　Mainly, I want to thank the students who have been in my classes over the years for making them so exciting. Many of you will recognize your own contributions, and I hope that you will forgive me for not naming you individually.
　　Acknowledgments for the First Edition
　　Several people used my notes and made valuable suggestions - Jay Goldman, Steve Kleiman, Richard Schafer, and Joe Silverman among them. Harold Stark helped me with the number theory, and Gil Strang with the linear algebra. Also, the following people read the manuscript and commented on it: Ellen Kirkman, Al Levine, Barbara Peskin, and John Tate. I want to thank Barbara Peskin especially for reading the whole thing twice during the final year.
　　The figures which needed mathematical precision were made on the computer by George Fann and Bill Schelter. I could not have done them by myself. Many thanks also to Marge Zabierek, who retyped the manuscript annually for about eight years before it was put onto the computer where I could do the revisions myself, and to Mary Roybal for her careful and expert job of editing the manuscript.
　　I haven't consulted other books very much while writing this one, but the classics by Birkhoff and MacLane and by van der Waerden from which I learned the subject influenced me a great deal, as did Herstein's book, which I used as a text for many years. I also found some good exercises in the books by Noble and by Paley and Weichsel.
　　Acknowledgments for the Second Edition
　　Many people have commented on the first edition - a few are mentioned in the text. I'm afraid that I will have forgotten to mention most of you.
　　I want to thank these people especially: Annette A' Campo and Paolo Maroscia made careful translations of the first edition, and gave me many corrections. Nathaniel Kuhn and James Lepowsky made valuable suggestions. Annette and Nat finally got through my thick skull how one should prove the orthogonality relations.
　　I thank the people who reviewed the manuscript for their suggestions. They include Alberto Corso, Thomas C. Craven, Sergi Elizade, Luis Finotti, Peter A. Linnell, Brad Shelton, Hema Srinivasan, and Nik Weaver. Toward the end of the process, Roger Lipsett read and commented on the entire revised manuscript. Brett Coonley helped with the many technical problems that arose when the manuscript was put into TeX.
　　Many thanks, also, to Caroline Celano at Pearson for her careful and thorough editing of the manuscript and to Patty Donovan at Laserwords, who always responded graciously to my requests for yet another emendation, though her patience must have been tried at times.And I talk to Gil Strang and Harold Stark often, about everything.
　　Finally, I want to thank the many MIT undergraduates who read and commented on the revised text and corrected errors. The readers include Nerses Aramyan, Reuben Aronson, Mark Chen, Jeremiah Edwards, Giuliano Giacaglia, Li-Mei L'un, Ana Malagon,Maria Monks, and Charmaine Sia. I came to rely heavily on them, especially on Nerses,Li-Mei, and Charmaine.
　　"One, two, three, five, four..."
　　"No Daddy, it's one, two, three, four, five."
　　"Well if I want to say one, two, three, five, four, why can't I?"
　　"That's not how it goes."
　　--Carolyn Artin