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Introduction to Algebra [Hardback]

  • Formāts: Hardback, 306 pages, height x width: 232x156 mm, weight: 605 g, 10 line figures
  • Izdošanas datums: 01-Jul-1998
  • Izdevniecība: Oxford University Press
  • ISBN-10: 0198501951
  • ISBN-13: 9780198501954
Citas grāmatas par šo tēmu:
Introduction to AlgebraPalielināt
  • Formāts: Hardback, 306 pages, height x width: 232x156 mm, weight: 605 g, 10 line figures
  • Izdošanas datums: 01-Jul-1998
  • Izdevniecība: Oxford University Press
  • ISBN-10: 0198501951
  • ISBN-13: 9780198501954
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780198501954.html
Keywords:
Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the theory of groups, rings and fields, coding theory, and Galois theory. With over 300 exercises, and web-based solutions, this is an ideal introductory text for Year 1 and 2 undergraduate students in mathematics.
1 Introduction
1(24)
1.1 What is algebra?
1(2)
1.2 Set theory
3(10)
1.2.1 Introduction
3(3)
1.2.2 Relations
6(3)
1.2.3 Functions and operations
9(2)
1.2.4 Logic and truth tables
11(2)
1.3 The integers
13(5)
1.4 Polynomials and matrices
18(7)
1.4.1 Polynomials
18(2)
1.4.2 Matrices
20(5)
2 Rings
25(40)
2.1 Rings and subrings
25(10)
2.1.1 Introduction
25(1)
2.1.2 Examples of rings
26(2)
2.1.3 Properties of rings
28(3)
2.1.4 Subrings
31(4)
2.2 Homomorphisms and ideals
35(12)
2.2.1 Cosets
35(1)
2.2.2 Homomorphisms and ideals
36(4)
2.2.3 Factor rings and isomorphism theorems
40(4)
2.2.4 Polynomials
44(3)
2.3 Factorisation
47(12)
2.3.1 Zero divisors and units
47(2)
2.3.2 Irreducibles and factorisation
49(2)
2.3.3 Principal ideal domains
51(3)
2.3.4 Euclidean domains
54(5)
2.4 Fields
59(4)
2.4.1 Field of fractions
59(1)
2.4.2 Maximal ideals and fields
60(1)
2.4.3 Field extensions, finite fields
61(2)
Appendix: Solution to Exercise 2.10
63(2)
3 Groups
65(38)
3.1 Groups and subgroups
65(9)
3.1.1 Introduction
65(1)
3.1.2 Examples of groups
66(3)
3.1.3 Properties of groups
69(2)
3.1.4 Subgroups
71(3)
3.2 Subgroups and cosets
74(4)
3.2.1 Cosets
74(1)
3.2.2 Orders; Lagrange's Theorem
75(1)
3.2.3 Cyclic groups
76(2)
3.3 Homomorphisms and normal subgroups
78(9)
3.3.1 Definitions
78(3)
3.3.2 Factor groups and isomorphism theorems
81(2)
3.3.3 Conjugacy
83(4)
3.4 Some special groups
87(14)
3.4.1 Cayley's Theorem
87(2)
3.4.2 Small groups
89(2)
3.4.3 Symmetric and alternating groups
91(4)
3.4.4 Symmetry groups
95(6)
Appendix: How many groups?
101(2)
4 Vector spaces
103(32)
4.1 Vector spaces and subspaces
103(11)
4.1.1 Introduction
103(2)
4.1.2 Examples
105(2)
4.1.3 Properties of vector spaces
107(1)
4.1.4 Subspaces
107(1)
4.1.5 Linear independence and bases
108(4)
4.1.6 Intersection and sum
112(2)
4.2 Linear transformations and matrices
114(21)
4.2.1 Linear transformations
114(3)
4.2.2 Matrices
117(1)
4.2.3 Change of basis
118(2)
4.2.4 Elementary operations
120(6)
4.2.5 Determinants
126(5)
4.2.6 Matrices over Euclidean domains
131(4)
5 Modules
135(24)
5.1 Introduction
135(8)
5.1.1 Definition of modules
135(2)
5.1.2 Examples of modules
137(2)
5.1.3 Submodules and homomorphisms
139(1)
5.1.4 Annihilators, cyclic modules, direct sums
139(4)
5.2 Modules over a Euclidean domain
143(6)
5.2.1 The structure theorem
143(3)
5.2.2 The primary decomposition
146(3)
5.3 Applications
149(10)
5.3.1 Finitely generated abelian groups
149(2)
5.3.2 Normal forms of matrices
151(3)
5.3.3 The Cayley-Hamilton Theorem
154(5)
6 The number systems
159(22)
6.1 To the complex numbers
160(10)
6.1.1 The natural numbers
160(2)
6.1.2 The integers
162(2)
6.1.3 The rational numbers
164(1)
6.1.4 The real numbers
165(3)
6.1.5 The complex numbers
168(2)
6.2 Algebraic and transcendental numbers
170(11)
6.2.1 Algebraic numbers
170(3)
6.2.2 Transcendental numbers
173(5)
6.2.3 Ruler-and-compass constructions
178(3)
7 Further topics
181(54)
7.1 Further group theory
181(14)
7.1.1 Permutation groups and group actions
181(2)
7.1.2 Sylow's Theorem
183(2)
7.1.3 The Jordan-Holder Theorem
185(1)
7.1.4 Soluble groups
186(2)
7.1.5 Simple groups
188(2)
7.1.6 Extensions
190(3)
7.1.7 A glimpse at homological algebra
193(2)
7.2 Further ring theory
195(9)
7.2.1 PID implies UFD
195(1)
7.2.2 Noetherian rings
196(2)
7.2.3 Gauss' Lemma
198(1)
7.2.4 Eisenstein's criterion
199(2)
7.2.5 A glimpse at algebraic geometry
201(3)
7.3 Further field theory
204(10)
7.3.1 Derivatives and repeated roots
204(3)
7.3.2 Splitting fields
207(2)
7.3.3 Finite fields
209(3)
7.3.4 Wedderburn's Theorem
212(2)
7.4 Other structures
214(21)
7.4.1 Universal algebra
214(4)
7.4.2 Lattices
218(6)
7.4.3 Category theory
224(11)
8 Applications
235(34)
8.1 Coding theory
235(18)
8.1.1 Codes
235(4)
8.1.2 Linear codes
239(2)
8.1.3 Syndrome decoding
241(3)
8.1.4 Cyclic codes
244(4)
8.1.5 BCH codes
248(5)
8.2 Galois theory
253(16)
8.2.1 Normality and separability
254(3)
8.2.2 The Main Theorem
257(2)
8.2.3 Solubility by radicals
259(4)
8.2.4 Ruler-and-compass revisited
263(2)
8.2.5 The Theorem of the Primitive Element
265(1)
Appendix: The Fundamental Theorem of Algebra
266(3)
Further reading 269(2)
Solutions to selected exercises 271(16)
Index 287