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El. knyga: Abstract Algebra

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This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven. The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader's appetite for further study. This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.
Preface xiii
Chapter 1 A Potpourri of Preliminary Topics
1(34)
1.1 What Are Definitions, Axioms, and Proofs?
1(1)
1.2 Mathematical Credos to Live By!
2(1)
1.3 A Smidgeon of Mathematical Logic and Some Proof Techniques
3(6)
1.4 A Smidgeon of Set Theory
9(3)
1.5 Functions
12(1)
1.6 Equivalence Relations
13(3)
1.7 Mathematical Induction
16(1)
1.8 A Smidgeon of Number Theory
17(4)
1.9 A Smidgeon of Combinatorics
21(14)
Exercises
26(9)
Chapter 2 Groups --- Part 1
35(28)
2.1 Introduction to Groups
35(4)
2.2 Abstract Groups
39(2)
2.3 Interesting Examples of Groups
41(2)
2.4 Group Homomorphisms
43(3)
2.5 Subgroups, Cosets, and Lagrange's Theorem
46(5)
2.6 Products of Groups
51(12)
Exercises
53(10)
Chapter 3 Rings --- Part 1
63(28)
3.1 Introduction to Rings
63(1)
3.2 Abstract Rings and Ring Homomorphisms
63(2)
3.3 Interesting Examples of Rings
65(4)
3.4 Some Important Special Types of Rings
69(1)
3.5 Unit Groups and Product Rings
70(2)
3.6 Ideals and Quotient Rings
72(5)
3.7 Prime Ideals and Maximal Ideals
77(14)
Exercises
79(12)
Chapter 4 Vector Spaces --- Part 1
91(14)
4.1 Introduction to Vector Spaces
91(1)
4.2 Vector Spaces and Linear Transformations
92(2)
4.3 Interesting Examples of Vector Spaces
94(1)
4.4 Bases and Dimension
95(10)
Exercises
100(5)
Chapter 5 Fields ---Part 1
105(22)
5.1 Introduction to Fields
105(1)
5.2 Abstract Fields and Homomorphisms
106(1)
5.3 Interesting Examples of Fields
107(1)
5.4 Subfields and Extension Fields
108(2)
5.5 Polynomial Rings
110(2)
5.6 Building Extension Fields
112(5)
5.7 Finite Fields
117(10)
Exercises
120(7)
Chapter 6 Groups --- Part 2
127(30)
6.1 Normal Subgroups and Quotient Groups
127(6)
6.2 Groups Acting on Sets
133(3)
6.3 The Orbit-Stabilizer Counting Theorem
136(4)
6.4 Sylow's Theorem
140(5)
6.5 Two Counting Lemmas
145(3)
6.6 Double Cosets and Sylow's Theorem
148(9)
Exercises
151(6)
Chapter 7 Rings --- Part 2
157(30)
7.1 Irreducible Elements and Unique Factorization Domains
157(2)
7.2 Euclidean Domains and Principal Ideal Domains
159(5)
7.3 Factorization in Principal Ideal Domains
164(3)
7.4 The Chinese Remainder Theorem
167(5)
7.5 Field of Fractions
172(4)
7.6 Multivariate and Symmetric Polynomials
176(11)
Exercises
180(7)
Chapter 8 Fields --- Part 2
187(34)
8.1 Algebraic Numbers and Transcendental Numbers
187(3)
8.2 Polynomial Roots and Multiplicative Subgroups
190(4)
8.3 Splitting Fields, Separability, and Irreducibility
194(6)
8.4 Finite Fields Revisited
200(1)
8.5 Gauss's Lemma and Eisenstein's Irreducibility Criterion
201(6)
8.6 Ruler and Compass Constructions
207(14)
Exercises
214(7)
Chapter 9 Galois Theory: Fields + Groups
221(74)
9.1 What Is Galois Theory?
221(1)
9.2 A Quick Review of Polynomials and Field Extensions
222(1)
9.3 Fields of Algebraic Numbers
223(3)
9.4 Algebraically Closed Fields
226(1)
9.5 Automorphisms of Fields
227(2)
9.6 Splitting Fields --- Part 1
229(6)
9.7 Splitting Fields --- Part 2
235(4)
9.8 The Primitive Element Theorem
239(3)
9.9 Galois Extensions
242(5)
9.10 The Fundamental Theorem of Galois Theory
247(3)
9.11 Application: The Fundamental Theorem of Algebra
250(4)
9.12 Galois Theory of Finite Fields
254(3)
9.13 A Plethora of Galois Equivalences
257(7)
9.14 Cyclotomic Fields and Kummer Fields
264(5)
9.15 Application: Insolubility of Polynomial Equations by Radicals
269(12)
9.16 Linear Independence of Field Automorphisms
281(14)
Exercises
285(10)
Chapter 10 Vector Spaces --- Part 2
295(32)
10.1 Vector Space Homomorphisms (aka Linear Transformations)
295(1)
10.2 Endomorphisms and Automorphisms
296(2)
10.3 Linear Transformations and Matrices
298(5)
10.4 Subspaces and Quotient Spaces
303(3)
10.5 Eigenvalues and Eigenvectors
306(3)
10.6 Determinants
309(6)
10.7 Determinants, Eigenvalues, and Characteristic Polynomials
315(3)
10.8 Inifinite-Dimensional Vector Spaces
318(9)
Exercises
320(7)
Chapter 11 Modules --- Part 1: Rings + Vector-Like Spaces
327(44)
11.1 What Is a Module?
327(1)
11.2 Examples of Modules
328(2)
11.3 Submodules and Quotient Modules
330(2)
11.4 Free Modules and Finitely Generated Modules
332(2)
11.5 Homomorphisms. Endomorphisms. Matrices
334(3)
11.6 Noetherian Rings and Modules
337(6)
11.7 Matrices with Entries in a Euclidean Domain
343(3)
11.8 Finitely Generated Modules over Euclidean Domains
346(7)
11.9 Applications of the Structure Theorem
353(18)
Exercises
357(14)
Chapter 12 Groups --- Part 3
371(26)
12.1 Permutation Groups
371(8)
12.2 Cayley's Theorem
379(1)
12.3 Simple Groups
380(6)
12.4 Composition Series
386(2)
12.5 Automorphism Groups
388(1)
12.6 Semidirect Products of Groups
389(3)
12.7 The Structure of Finite Abelian Groups
392(5)
Exercises
393(4)
Chapter 13 Modules --- Part 2: Multilinear Algebra
397(16)
13.1 Multilinear Maps and Multilinear Forms
397(2)
13.2 Symmetric and Alternating Forms
399(2)
13.3 Alternating Forms on Free Modules
401(4)
13.4 The Determinant Map
405(8)
Exercises
409(4)
Chapter 14 Additional Topics in Brief
413(110)
14.1 Sets Countable and Uncountable
413(4)
14.2 The Axiom of Choice
417(5)
14.3 Tensor Products and Multilinear Algebra
422(4)
14.4 Commutative Algebra
426(9)
14.5 Category Theory
435(8)
14.6 Graph Theory
443(5)
14.7 Representation Theory
448(9)
14.8 Elliptic Curves
457(7)
14.9 Algebraic Number Theory
464(6)
14.10 Algebraic Geometry
470(7)
14.11 Euclidean Lattices
477(12)
14.12 Non-Commutative Rings
489(7)
14.13 Mathematical Cryptography
496(27)
Exercises
504(19)
Sample Syllabi 523(4)
List of Notation 527(6)
List of Figures 533(4)
Index 537
Joseph H. Silverman, Brown University, Providence, RI
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