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Survey of Modern Algebra [Hardback]

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  • Formāts: Hardback, 512 pages, height x width: 229x152 mm, weight: 1110 g
  • Izdošanas datums: 06-Feb-1998
  • Izdevniecība: A K Peters
  • ISBN-10: 1568810687
  • ISBN-13: 9781568810683
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  • Cena: 119,73 €
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Survey of Modern AlgebraPalielināt
  • Formāts: Hardback, 512 pages, height x width: 229x152 mm, weight: 1110 g
  • Izdošanas datums: 06-Feb-1998
  • Izdevniecība: A K Peters
  • ISBN-10: 1568810687
  • ISBN-13: 9781568810683
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781568810683.html
A new edition of the classic undergraduate text introducing abstract algebra using concrete examples. The authors ground their explanations with computational and theoretical exercises to develop the student's "power to think for himself," covering topics such as the role of careful proof in algebra, linear algebra as grounded in geometry, groups as expressions of symmetry, and subgroups and subsystems leading to lattice theory. This volume is a corrected version of the 4th edition and is offered by a new publisher. Annotation c. by Book News, Inc., Portland, Or.
Preface to the Fourth Edition v
1 The Integers
1(37)
1.1 Commutative Rings; Integral Domains
1(2)
1.2 Elementary Properties of Commutative Rings
3(5)
1.3 Ordered Domains
8(3)
1.4 Well-Ordering Principle
11(1)
1.5 Finite Induction; Laws of Exponents
12(4)
1.6 Divisibility
16(2)
1.7 The Euclidean Algorithm
18(5)
1.8 Fundamental Theorem of Arithmetic
23(2)
1.9 Congruences
25(4)
1.10 The Rings Zn
29(3)
1.11 Sets, Functions, and Relations
32(3)
1.12 Isomorphisms and Automorphisms
35(3)
2 Rational Numbers and Fields
38(23)
2.1 Definition of a Field
38(4)
2.2 Construction of the Rationals
42(5)
2.3 Simultaneous Linear Equations
47(5)
2.4 Ordered Fields
52(2)
2.5 Postulates for the Positive Integers
54(3)
2.6 Peano Postulates
57(4)
3 Polynomials
61(33)
3.1 Polynomial Forms
61(4)
3.2 Polynomial Functions
65(4)
3.3 Homomorphisms of Commutative Rings
69(3)
3.4 Polynomials in Several Variables
72(2)
3.5 The Division Algorithm
74(2)
3.6 Units and Associates
76(2)
3.7 Irreducible Polynomials
78(2)
3.8 Unique Factorization Theorem
80(4)
3.9 Other Domains with Unique Factorization
84(4)
3.10 Eisenstein's Irreducibility Criterion
88(2)
3.11 Partial Fractions
90(4)
4 Real Numbers
94(13)
4.1 Dilemma of Pythagoras
94(2)
4.2 Upper and Lower Bounds
96(2)
4.3 Postulates for Real Numbers
98(3)
4.4 Roots of Polynomial Equations
101(3)
4.5 Dedekind Cuts
104(3)
5 Complex Numbers
107(17)
5.1 Definition
107(3)
5.2 The Complex Plane
110(3)
5.3 Fundamental Theorem of Algebra
113(4)
5.4 Conjugate Numbers and Real Polynomials
117(1)
5.5 Quadratic and Cubic Equations
118(3)
5.6 Solution of Quartic by Radicals
121(1)
5.7 Equations of Stable Type
122(3)
6 Groups
124(44)
6.1 Symmetries of the Square
124(2)
6.2 Groups of Transformations
126(5)
6.3 Further Examples
131(2)
6.4 Abstract Groups
133(4)
6.5 Isomorphism
137(3)
6.6 Cyclic Groups
140(3)
6.7 Subgroups
143(3)
6.8 Lagrange's Theorem
146(4)
6.9 Permutation Groups
150(3)
6.10 Even and Odd Permutations
153(2)
6.11 Homomorphisms
155(2)
6.12 Automorphisms; Conjugate Elements
157(4)
6.13 Quotient Groups
161(3)
6.14 Equivalence and Congruence Relations
164(4)
7 Vectors and Vector Spaces
168(46)
7.1 Vectors in a Plane
168(1)
7.2 Generalizations
169(2)
7.3 Vector Spaces and Subspaces
171(5)
7.4 Linear Independence and Dimension
176(4)
7.5 Matrices and Row-equivalence
180(3)
7.6 Tests for Linear Dependence
183(5)
7.7 Vector Equations; Homogeneous Equations
188(5)
7.8 Bases and Coordinate Systems
193(5)
7.9 Inner Products
198(2)
7.10 Euclidean Vector Spaces
200(3)
7.11 Normal Orthogonal Bases
203(3)
7.12 Quotient-spaces
206(2)
7.13 Linear Functions and Dual Spaces
208(6)
8 The Algebra of Matrices
214(46)
8.1 Linear Transformations and Matrices
214(6)
8.2 Matrix Addition
220(2)
8.3 Matrix Multiplication
222(6)
8.4 Diagonal, Permutation, and Triangular Matrices
228(2)
8.5 Rectangular Matrices
230(5)
8.6 Inverses
235(6)
8.7 Rank and Nullity
241(2)
8.8 Elementary Matrices
243(5)
8.9 Equivalence and Canonical Form
248(3)
8.10 Bilinear Functions and Tensor Products
251(4)
8.11 Quaternions
255(5)
9 Linear Groups
260(58)
9.1 Change of Basis
260(3)
9.2 Similar Matrices and Eigenvectors
263(5)
9.3 The Full Linear and Affine Groups
268(4)
9.4 The Orthogonal and Euclidean Groups
272(5)
9.5 Invariants and Canonical Forms
277(3)
9.6 Linear and Bilinear Forms
280(3)
9.7 Quadratic Forms
283(3)
9.8 Quadratic Forms Under the Full Linear Group
286(2)
9.9 Real Quadratic Forms Under the Full Linear Group
288(4)
9.10 Quadratic Forms Under the Orthogonal Group
292(4)
9.11 Quadrics Under the Affine and Euclidean Groups
296(4)
9.12 Unitary and Hermitian Matrices
300(5)
9.13 Affine Geometry
305(7)
9.14 Projective Geometry
312(6)
10 Determinants and Canonical Forms
318(39)
10.1 Definition and Elementary Properties of Determinants
318(5)
10.2 Products of Determinants
323(4)
10.3 Determinants as Volumes
327(4)
10.4 The Characteristic Polynomial
331(5)
10.5 The Minimal Polynomial
336(4)
10.6 Cayley-Hamilton Theorem
340(2)
10.7 Invariant Subspaces and Reducibility
342(4)
10.8 First Decomposition Theorem
349(3)
10.9 Second Decomposition Theorem
349(3)
10.10 Rational and Jordan Canonical Forms
352(5)
11 Boolean Algebras and Lattices
357(24)
11.1 Basic Definition
357(2)
11.2 Laws: Analogy with Airthmetic
359(2)
11.3 Boolean Algebra
361(3)
11.4 Deduction of Other Basic Laws
364(4)
11.5 Canonical Forms of Boolean Polynomials
368(3)
11.6 Partial Orderings
371(3)
11.7 Lattices
374(3)
11.8 Representation by Sets
377(4)
12 Transfinite Arithmetic
381(14)
12.1 Numbers and Sets
381(2)
12.2 Countable Sets
383(3)
12.3 Other Cardinal Numbers
386(4)
12.4 Addition and Multiplication of Cardinals
390(2)
12.5 Exponentiation
392(3)
13 Rings and Ideals
395(25)
13.1 Rings
395(4)
13.2 Homomorphisms
399(4)
13.3 Quotient-rings
403(4)
13.4 Algebra of Ideals
407(3)
13.5 Polynomial Ideals
410(3)
13.6 Ideals in Linear Algebras
413(2)
13.7 The Characteristic of a Ring
415(3)
13.8 Characteristics of Fields
418(2)
14 Algebraic Number Fields
420(32)
14.1 Algebraic and Transcendental Extensions
420(3)
14.2 Elements Algebraic over a Field
423(2)
14.3 Adjunction of Roots
425(4)
14.4 Degrees and Finite Extensions
429(2)
14.5 Iterated Algebraic Extensions
431(4)
14.6 Algebraic Numbers
435(4)
14.7 Gaussian Integers
439(4)
14.8 Algebraic Integers
443(2)
14.9 Sums and Products of Integers
445(3)
14.10 Factorization of Quadratic Integers
448(4)
15 Galois Theory
452(31)
15.1 Root Fields for Equations
452(2)
15.2 Uniqueness Theorem
454(2)
15.3 Finite Fields
456(3)
15.4 The Galois Group
459(5)
15.5 Separable and Inseparable Polynomials
464(3)
15.6 Properties of the Galois Group
467(4)
15.7 Subgroups and Subfields
471(3)
15.8 Irreducible Cubic Equations
474(4)
15.9 Insolvability of Quintic Equations
478(5)
Bibliography 483(3)
List of Special Symbols 486(3)
Index 489


Birkhoff , Garrett; Mac Lane , Saunders