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E-raamat: Algebra: A Teaching and Source Book

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  • Ilmumisaeg: 14-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319197340
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Algebra: A Teaching and Source BookSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 14-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319197340
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This book presents a graduate-level course on modern algebra. It can be used as a teaching book – owing to the copious exercises – and as a source book for those who wish to use the major theorems of algebra.

The course begins with the basic combinatorial principles of algebra: posets, chain conditions, Galois connections, and dependence theories. Here, the general Jordan–Holder Theorem becomes a theorem on interval measures of certain lower semilattices. This is followed by basic courses on groups, rings and modules; the arithmetic of integral domains; fields; the categorical point of view; and tensor products.

Beginning with introductory concepts and examples, each chapter proceeds gradually towards its more complex theorems. Proofs progress step-by-step from first principles. Many interesting results reside in the exercises, for example, the proof that ideals in a Dedekind domain are generated by at most two elements. The emphasis throughout is on real understanding as opposed to memorizing a catechism and so some chapters offer curiosity-driven appendices for the self-motivated student.

Arvustused

While this book requires a level of mathematical maturity that can reasonably be expected from a student entering graduate study, it does not assume too muchand that may be its greatest strength. Exercises are interesting and challenging. The work is well suited for a full-year graduate course in algebra. The benefit of years of teaching by both authors is evident in the style and exposition. (R. J. Bumcrot, Mathematical Reviews, February, 2016)

This book covers the fundamental concepts and results in what is generally labelled as abstract, higher or modern algebra at a graduate level. Due to an impressive amount of material presented in a very articulated manner, this book is a valuable addition to the literature. One of the strong points of the book is the information shared with readers in footnotes offering a broader perspective onthe topic. (Mihai Cipu, zbMATH 1346.00003, 2016)

This is a text for a one-year course in abstract algebra at the first-year graduate level. this is an interesting take on graduate algebra, and anybody who teaches a course on that subject should certainly take a look at this book. (Mark Hunacek, MAA Reviews, September, 2015)

1 Basics
1(20)
1.1 Presumed Results and Conventions
1(11)
1.1.1 Presumed Jargon
1(1)
1.1.2 Basic Arithmetic
2(2)
1.1.3 Sets and Maps
4(6)
1.1.4 Notation for Compositions of Mappings
10(2)
1.2 Binary Operations and Monoids
12(2)
1.3 Notation for Special Structures
14(1)
1.4 The Axiom of Choice and Cardinal Numbers
15(6)
1.4.1 The Axiom of Choice
15(1)
1.4.2 Cardinal Numbers
15(4)
References
19(2)
2 Basic Combinatorial Principles of Algebra
21(52)
2.1 Introduction
21(1)
2.2 Basic Definitions
22(19)
2.2.1 Definition of a Partially Ordered Set
22(1)
2.2.2 Subposets and Induced Subposets
23(1)
2.2.3 Dual Posets and Dual Concepts
24(1)
2.2.4 Maximal and Minimal Elements of Induced Posets
24(1)
2.2.5 Global Maxima and Minima
25(1)
2.2.6 Total Orderings and Chains
25(1)
2.2.7 Zornification
26(1)
2.2.8 Well-Ordered Sets
26(4)
2.2.9 Order Ideals and Filters
30(2)
2.2.10 Antichains
32(1)
2.2.11 Products of Posets
32(1)
2.2.12 Morphisms of Posets
33(1)
2.2.13 Examples
34(3)
2.2.14 Closure Operators
37(1)
2.2.15 Closure Operators and Galois Connections
38(3)
2.3 Chain Conditions
41(4)
2.3.1 Saturated Chains
41(1)
2.3.2 Algebraic Intervals and the Height Function
41(1)
2.3.3 The Ascending and Descending Chain Conditions in Arbitrary Posets
42(3)
2.4 Posets with Meets and/or Joins
45(4)
2.4.1 Meets and Joins
45(2)
2.4.2 Lower Semilattices with the Descending Chain Condition
47(1)
2.4.3 Lower Semilattices with both Chain Conditions
48(1)
2.5 The Jordan-Holder Theory
49(10)
2.5.1 Lower Semi-lattices and Semimodularity
50(1)
2.5.2 Interval Measures on Posets
51(1)
2.5.3 The Jordan-Holder Theorem
52(1)
2.5.4 Modular Lattices
53(6)
2.6 Dependence Theories
59(7)
2.6.1 Introduction
59(1)
2.6.2 Dependence
59(1)
2.6.3 Extending the Definition to S × P(S)
60(1)
2.6.4 Independence and Spanning
61(1)
2.6.5 Dimension
62(1)
2.6.6 Other Formulations of Dependence Theory
63(3)
2.7 Exercises
66(7)
2.7.1 Exercises for Sect. 2.2
66(2)
2.7.2 Exercises for Sects. 2.3 and 2.4
68(2)
2.7.3 Exercises for Sect. 2.5
70(1)
2.7.4 Exercises for Sect. 2.6
71(1)
References
71(2)
3 Review of Elementary Group Properties
73(32)
3.1 Introduction
73(1)
3.2 Definition and Examples
74(6)
3.2.1 Orders of Elements and Groups
77(1)
3.2.2 Subgroups
77(2)
3.2.3 Cosets, and Lagrange's Theorem in Finite Groups
79(1)
3.3 Homomorphisms of Groups
80(9)
3.3.1 Definitions and Basic Properties
80(1)
3.3.2 Automorphisms as Right Operators
81(1)
3.3.3 Examples of Homomorphisms
82(7)
3.4 Factor Groups and the Fundamental Theorems of Homomorphisms
89(5)
3.4.1 Introduction
89(1)
3.4.2 Normal Subgroups
89(2)
3.4.3 Factor Groups
91(2)
3.4.4 Normalizers and Centralizers
93(1)
3.5 Direct Products
94(2)
3.6 Other Variations: Semidirect Products and Subdirect Products
96(5)
3.6.1 Semidirect Products
96(3)
3.6.2 Subdirect Products
99(2)
3.7 Exercises
101(4)
3.7.1 Exercises for Sects. 3.1--3.3
101(1)
3.7.2 Exercises for Sect. 3.4
101(1)
3.7.3 Exercises for Sects. 3.5--3.7
102(3)
4 Permutation Groups and Group Actions
105(32)
4.1 Group Actions and Orbits
105(2)
4.1.1 Examples of Group Actions
106(1)
4.2 Permutations
107(9)
4.2.1 Cycle Notation
108(1)
4.2.2 Even and Odd Permutations
109(2)
4.2.3 Transpositions and Cycles: The Theorem of Feit, Lyndon and Scott
111(2)
4.2.4 Action on Right Cosets
113(1)
4.2.5 Equivalent Actions
114(1)
4.2.6 The Fundamental Theorem of Transitive Actions
114(1)
4.2.7 Normal Subgroups of Transitive Groups
115(1)
4.2.8 Double Cosets
116(1)
4.3 Applications of Transitive Group Actions
116(7)
4.3.1 Cardinalities of Conjugacy Classes of Subsets
116(1)
4.3.2 Finite p-groups
117(1)
4.3.3 The Sylow Theorems
117(2)
4.3.4 Fusion and Transfer
119(2)
4.3.5 Calculations of Group Orders
121(2)
4.4 Primitive and Multiply Transitive Actions
123(8)
4.4.1 Primitivity
123(1)
4.4.2 The Rank of a Group Action
124(3)
4.4.3 Multiply Transitive Group Actions
127(3)
4.4.4 Permutation Characters
130(1)
4.5 Exercises
131(6)
4.5.1 Exercises for Sects. 4.1--42
131(2)
4.5.2 Exercises Involving Sects. 4.3 and 4.4
133(4)
5 Normal Structure of Groups
137(26)
5.1 The Jordan-Holder Theorem for Artinian Groups
137(3)
5.1.1 Subnormality
137(3)
5.2 Commutators
140(2)
5.3 The Derived Series and Solvability
142(4)
5.4 Central Series and Nilpotent Groups
146(6)
5.4.1 The Upper and Lower Central Series
146(3)
5.4.2 Finite Nilpotent Groups
149(3)
5.5 Coprime Action
152(4)
5.6 Exercises
156(7)
5.6.1 Elementary Exercises
156(3)
5.6.2 The Baer-Suzuki Theorem
159(4)
6 Generation in Groups
163(22)
6.1 Introduction
163(1)
6.2 The Cayley Graph
164(1)
6.2.1 Definition
164(1)
6.2.2 Morphisms of Various Sorts of Graphs
164(1)
6.2.3 Group Morphisms Induce Morphisms of Cayley Graphs
165(1)
6.3 Free Groups
165(13)
6.3.1 Construction of Free Groups
165(6)
6.3.2 The Universal Property
171(1)
6.3.3 Generators and Relations
171(7)
6.4 The Brauer-Ree Theorem
178(2)
6.5 Exercises
180(2)
6.5.1 Exercises for Sect. 6.3
180(2)
6.5.2 Exercises for Sect. 6.4
182(1)
6.6 A Word to the Student
182(3)
6.6.1 The Classification of Finite Simple Groups
182(1)
References
183(2)
7 Elementary Properties of Rings
185(46)
7.1 Elementary Facts About Rings
185(4)
7.1.1 Introduction
185(1)
7.1.2 Definitions
185(3)
7.1.3 Units in Rings
188(1)
7.2 Homomorphisms
189(9)
7.2.1 Ideals and Factor Rings
191(1)
7.2.2 The Fundamental Theorems of Ring Homomorphisms
192(1)
7.2.3 The Poset of Ideals
193(5)
7.3 Monoid Rings and Polynomial Rings
198(12)
7.3.1 Some Classic Monoids
198(1)
7.3.2 General Monoid Rings
199(3)
7.3.3 Group Rings
202(1)
7.3.4 Mobius Algebras
202(1)
7.3.5 Polynomial Rings
202(6)
7.3.6 Algebraic Varieties: An Application of the Polynomial Rings F[ X]
208(2)
7.4 Other Examples and Constructions of Rings
210(12)
7.4.1 Examples
210(8)
7.4.2 Integral Domains
218(4)
7.5 Exercises
222(9)
7.5.1 Warmup Exercises for Sect. 7.1
222(1)
7.5.2 Exercises for Sect. 7.2
223(2)
7.5.3 Exercises for Sect. 7.3
225(4)
7.5.4 Exercises for Sect. 7.4
229(1)
References
230(1)
8 Elementary Properties of Modules
231(48)
8.1 Basic Theory
231(15)
8.1.1 Modules over Rings
231(1)
8.1.2 The Connection Between R-Modules and Endomorphism Rings of Additive Groups
232(2)
8.1.3 Bimodules
234(1)
8.1.4 Submodules
235(1)
8.1.5 Factor Modules
236(1)
8.1.6 The Fundamental Homomorphism Theorems for R-Modules
237(2)
8.1.7 The Jordan-Holder Theorem for R-Modules
239(2)
8.1.8 Direct Sums and Products of Modules
241(2)
8.1.9 Free Modules
243(2)
8.1.10 Vector Spaces
245(1)
8.2 Consequences of Chain Conditions on Modules
246(10)
8.2.1 Noetherian and Artinian Modules
246(2)
8.2.2 Effects of the Chain Conditions on Endomorphisms
248(5)
8.2.3 Noetherian Modules and Finite Generation
253(1)
8.2.4 E. Noether's Theorem
253(1)
8.2.5 Algebraic Integers and Integral Elements
254(2)
8.3 The Hilbert Basis Theorem
256(2)
8.4 Mapping Properties of Modules
258(13)
8.4.1 The Universal Mapping Properties of the Direct Sum and Direct Product
258(3)
8.4.2 HomR(M, N)
261(2)
8.4.3 Exact Sequences
263(2)
8.4.4 Projective and Injective Modules
265(6)
8.5 Exercises
271(8)
8.5.1 Exercises for Sect. 8.1
271(2)
8.5.2 Exercises for Sects. 8.2 and 8.3
273(1)
8.5.3 Exercises for Sect. 8.4
274(3)
Reference
277(2)
9 The Arithmetic of Integral Domains
279(54)
9.1 Introduction
279(1)
9.2 Divisibility and Factorization
280(5)
9.2.1 Divisibility
280(3)
9.2.2 Factorization and Irreducible Elements
283(1)
9.2.3 Prime Elements
284(1)
9.3 Euclidean Domains
285(4)
9.3.1 Examples of Euclidean Domains
287(2)
9.4 Unique Factorization
289(3)
9.4.1 Factorization into Prime Elements
289(3)
9.4.2 Principal Ideal Domains Are Unique Factorization Domains
292(1)
9.5 If D Is a UFD, Then so Is D[ x]
292(4)
9.6 Localization in Commutative Rings
296(4)
9.6.1 Localization by a Multiplicative Set
296(1)
9.6.2 Special Features of Localization in Domains
297(1)
9.6.3 A Local Characterization of UFD's
298(2)
9.6.4 Localization at a Prime Ideal
300(1)
9.7 Integral Elements in a Domain
300(2)
9.7.1 Introduction
300(2)
9.8 Rings of Integral Elements
302(3)
9.9 Factorization Theory in Dedekind Domains and the Fundamental Theorem of Algebraic Number Theory
305(4)
9.10 The Ideal Class Group of a Dedekind Domain
309(1)
9.11 A Characterization of Dedekind Domains
310(4)
9.12 When Are Rings of Integers Dedekind?
314(3)
9.13 Exercises
317(16)
9.13.1 Exercises for Sects. 9.2, 9.3, 9.4 and 9.5
317(3)
9.13.2 Exercises on Localization
320(1)
9.13.3 Exercises for Sect. 9.9
321(1)
9.13.4 Exercises for Sect. 9.10
322(1)
9.13.5 Exercises for Sect. 9.11
323(1)
9.13.6 Exercises for Sect. 9.12
324(1)
Appendix: The Arithmetic of Quadratic Domains
325(8)
10 Principal Ideal Domains and Their Modules
333(22)
10.1 Introduction
333(1)
10.2 Quick Review of PID's
334(1)
10.3 Free Modules over a PID
335(1)
10.4 A Technical Result
335(3)
10.5 Finitely Generated Modules over a PID
338(4)
10.6 The Uniqueness of the Invariant Factors
342(1)
10.7 Applications of the Theorem on PID Modules
343(8)
10.7.1 Classification of Finite Abelian Groups
343(1)
10.7.2 The Rational Canonical Form of a Linear Transformation
344(4)
10.7.3 The Jordan Form
348(2)
10.7.4 Information Carried by the Invariant Factors
350(1)
10.8 Exercises
351(4)
10.8.1 Miscellaneous Exercises
351(2)
10.8.2 Structure Theory of Modules over Z and Polynomial Rings
353(2)
11 Theory of Fields
355(88)
11.1 Introduction
355(1)
11.2 Elementary Properties of Field Extensions
356(7)
11.2.1 Algebraic and Transcendental Extensions
357(1)
11.2.2 Indices Multiply: Ruler and Compass Problems
358(2)
11.2.3 The Number of Zeros of a Polynomial
360(3)
11.3 Splitting Fields and Their Automorphisms
363(7)
11.3.1 Extending Isomorphisms Between Fields
363(1)
11.3.2 Splitting Fields
364(4)
11.3.3 Normal Extensions
368(2)
11.4 Some Applications to Finite Fields
370(4)
11.4.1 Automorphisms of Finite Fields
370(2)
11.4.2 Polynomial Equations Over Finite Fields: The Chevalley-Warning Theorem
372(2)
11.5 Separable Elements and Field Extensions
374(6)
11.5.1 Separability
374(3)
11.5.2 Separable and Inseparable Extensions
377(3)
11.6 Galois Theory
380(7)
11.6.1 Galois Field Extensions
380(1)
11.6.2 The Dedekind Independence Lemma
381(3)
11.6.3 Galois Extensions and the Fundamental Theorem of Galois Theory
384(3)
11.7 Traces and Norms and Their Applications
387(4)
11.7.1 Introduction
387(1)
11.7.2 Elementary Properties of the Trace and Norm
388(1)
11.7.3 The Trace Map for a Separable Extension Is Not Zero
389(1)
11.7.4 An Application to Rings of Integral Elements
390(1)
11.8 The Galois Group of a Polynomial
391(7)
11.8.1 The Cyclotomic Polynomials
391(1)
11.8.2 The Galois Group as a Permutation Group
392(6)
11.9 Solvability of Equations by Radicals
398(8)
11.9.1 Introduction
398(3)
11.9.2 Roots of Unity
401(1)
11.9.3 Radical Extensions
402(1)
11.9.4 Galois' Criterion for Solvability by Radicals
403(3)
11.10 The Primitive Element Theorem
406(1)
11.11 Transcendental Extensions
407(7)
11.11.1 Simple Transcendental Extensions
407(7)
11.12 Exercises
414(29)
11.12.1 Exercises for Sect. 11.2
414(2)
11.12.2 Exercises for Sect. 11.3
416(1)
11.12.3 Exercises for Sect. 11.4
416(2)
11.12.4 Exercises for Sect. 11.5
418(1)
11.12.5 Exercises for Sect. 11.6
418(1)
11.12.6 Exercises for Sect. 11.8
419(2)
11.12.7 Exercises for Sects. 11.9 and 11.10
421(2)
11.12.8 Exercises for Sect. 11.11
423(1)
11.12.9 Exercises Associated with Appendix 1 of Chap. 10
424(1)
Appendix 1 Fields with a Valuation
424(17)
Reference
441(2)
12 Semiprime Rings
443(28)
12.1 Introduction
443(1)
12.2 Complete Reducibility
444(3)
12.3 Homogeneous Components
447(2)
12.3.1 The Action of the R-Endomorphism Ring
447(1)
12.3.2 The Socle of a Module Is a Direct Sum of the Homogeneous Components
448(1)
12.4 Semiprime Rings
449(6)
12.4.1 Introduction
449(1)
12.4.2 The Semiprime Condition
450(1)
12.4.3 Completely Reducible and Semiprimitive Rings Are Species of Semiprime Rings
451(1)
12.4.4 Idempotents and Minimal Right Ideals of Semiprime Rings
452(3)
12.5 Completely Reducible Rings
455(6)
12.6 Exercises
461(10)
12.6.1 General Exercises
461(2)
12.6.2 Warm Up Exercises for Sects. 12.1 and 12.2
463(1)
12.6.3 Exercises Concerning the Jacobson Radical
464(1)
12.6.4 The Jacobson Radical of Artinian Rings
465(1)
12.6.5 Quasiregularity and the Radical
466(1)
12.6.6 Exercises Involving Nil One-Sided Ideals in Noetherian Rings
467(2)
References
469(2)
13 Tensor Products
471(58)
13.1 Introduction
471(1)
13.2 Categories and Universal Constructions
472(7)
13.2.1 Examples of Universal Constructions
472(1)
13.2.2 Definition of a Category
473(3)
13.2.3 Initial and Terminal Objects, and Universal Mapping Properties
476(1)
13.2.4 Opposite Categories
477(1)
13.2.5 Functors
478(1)
13.3 The Tensor Product as an Abelian Group
479(9)
13.3.1 The Defining Mapping Property
479(2)
13.3.2 Existence of the Tensor Product
481(1)
13.3.3 Mapping Properties of the Tensor Product
482(5)
13.3.4 The Adjointness Relationship of the Horn and Tensor Functors
487(1)
13.4 The Tensor Product as a Right S-Module
488(3)
13.5 Multiple Tensor Products and R-Multilinear Maps
491(2)
13.6 Interlude: Algebras
493(3)
13.7 The Tensor Product of R-Algebras
496(1)
13.8 Graded Algebras
497(4)
13.9 The Tensor Algebras
501(3)
13.9.1 Introduction
501(1)
13.9.2 The Tensor Algebra: As a Universal Mapping Property
501(1)
13.9.3 The Tensor Algebra: The Standard Construction
502(2)
13.10 The Symmetric and Exterior Algebras
504(8)
13.10.1 Definitions of the Algebras
504(2)
13.10.2 Applications of Theorem 13.10.1
506(6)
13.11 Basic Multilinear Algebra
512(2)
13.12 Last Words
514(1)
13.13 Exercises
515(14)
13.13.1 Exercises for Sect. 13.2
515(1)
13.13.2 Exercises for Sect. 13.3
516(1)
13.13.3 Exercises for Sect. 13.3.4
517(1)
13.13.4 Exercises for Sect. 13.4
518(3)
13.13.5 Exercises for Sects. 13.6 and 13.7
521(1)
13.13.6 Exercises for Sect. 13.8
522(1)
13.13.7 Exercises for Sect. 13.10
522(5)
13.13.8 Exercise for Sect. 13.11
527(1)
References
527(2)
Bibliography 529(2)
Index 531
Ernest Shult studied finite groups with Michio Suzuki and held visiting fellowships at the University of Chicago and the Princeton Institute for Advanced Study in the 1960's. He continued to contribute to finite groups until he got interested in incidence geometry, In 1987-8 he received a US Scientist Award from the Alexander von Humboldt Foundation in Freiburg Germany.

David Surowski studied with Larry C. Grove and wrote numerous papers on the representation theory of groups with (B,N) pairs. Eventually his research came to include regular maps on surfaces. He was the recipient of many teaching awards and directed two summer institutes for young students of high ability. He died in March 2011.
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