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Searching... | 30000003503731 | QA162 S62 2009 | Open Access Book | Book | Searching... |
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Summary
Summary
Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles.
A Quick Introduction to Algebra
The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level.
The Basics of Abstract Algebra for a First-Semester Course
Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange's theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley's theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields.
Advanced Topics for a Second-Semester Course
The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups.
Reviews 1
Choice Review
Smith's update to the first edition (CH, Jul'09, 46-6260) is an alternative approach to the usual first semester in higher algebra. The author accomplishes this by including many topics often absent from a first course, such as quasigroups, Noetherian domains, and modules, which, theoretically, are developed alongside their mainstream analogues, like groups, rings, and vector spaces. It is essentially a first semester wink at universal algebra. Smith's approach to axiomatic systems is few-too-many--he starts with structures with very few axioms, like semigroups and monoids, and continues adding axioms. He finishes with more complex axiomatic systems, like unique factorization domains and fields. The book is very well written and easy to read, flowing naturally from one topic to the next. Numerous supportive homework exercises are also included to help the reader explore further topics. This book will best serve readers with a background in abstract algebra who desire to strengthen their understanding and build bridges between various topics. Unfortunately, because many similar topics are handled in tandem, an inexperienced reader might become confused, especially as many clarifying examples are missing. This book is for readers who want an under the hood view of algebra. Summing Up: Recommended. Lower-division undergraduates and above; researchers and faculty --Andrew Misseldine, Southern Utah University
Table of Contents
1 Numbers | p. 1 |
1.1 Ordering numbers | p. 1 |
1.2 The Well-Ordering Principle | p. 3 |
1.3 Divisibility | p. 5 |
1.4 The Division Algorithm | p. 6 |
1.5 Greatest common divisors | p. 9 |
1.6 The Euclidean Algorithm | p. 10 |
1.7 Primes and irreducibles | p. 13 |
1.8 The Fundamental Theorem of Arithmetic | |
1.9 Exercises | p. 17 |
1.10 Study projects | p. 22 |
1.11 Notes | p. 23 |
2 Functions | p. 25 |
2.1 Specifying functions | p. 25 |
2.2 Composite functions | p. 27 |
2.3 Linear functions | p. 28 |
2.4 Semigroups of functions | p. 29 |
2.5 Injectivity and surjectivity | p. 31 |
2.6 Isomorphisms | p. 34 |
2.7 Groups of permutations | p. 36 |
2.8 Exercises | p. 39 |
2.9 Study projects | p. 43 |
2.10 Notes | p. 46 |
2.11 Summary | p. 47 |
3 Equivalence | p. 49 |
3.1 Kernel and equivalence relations | p. 49 |
3.2 Equivalence classes | p. 51 |
3.3 Rational numbers | p. 53 |
3.4 The First Isomorphism Theorem for Sets | p. 56 |
3.5 Modular arithmetic | p. 58 |
3.6 Exercises | p. 61 |
3.7 Study projects | p. 63 |
3.8 Notes | p. 66 |
4 Groups and Monoids | p. 67 |
4.1 Semigroups | p. 67 |
4.2 Monoids | p. 69 |
4.3 Groups | p. 71 |
4.4 Componentwise structure | p. 73 |
4.5 Powers | p. 77 |
4.6 Submonoids and subgroups | p. 78 |
4.7 Cosets | p. 82 |
4.8 Multiplication tables | p. 84 |
4.9 Exercises | p. 87 |
4.10 Study projects | p. 91 |
4.11 Notes | p. 94 |
5 Homomorphisms | p. 95 |
5.1 Homomorphisms | p. 95 |
5.2 Normal subgroups | p. 98 |
5.3 Quotients | p. 101 |
5.4 The First Isomorphism Theorem for Groups | p. 104 |
5.5 The Law of Exponents | p. 106 |
5.6 Cayley's Theorem | p. 109 |
5.7 Exercises | p. 112 |
5.8 Study projects | p. 116 |
5.9 Notes | p. 125 |
6 Rings | p. 127 |
6.1 Rings | p. 127 |
6.2 Distributivity | p. 131 |
6.3 Subrings | p. 133 |
6.4 Ring homomorphisms | p. 135 |
6.5 Ideals | p. 137 |
6.6 Quotient rings | p. 139 |
6.7 Polynomial rings | p. 140 |
6.8 Substitution | p. 145 |
6.9 Exercises | p. 147 |
6.10 Study projects | p. 151 |
6.11 Notes | p. 156 |
7 Fields | p. 157 |
7.1 Integral domains | p. 157 |
7.2 Degrees | p. 160 |
7.3 Fields | p. 162 |
7.4 Polynomials over fields | p. 164 |
7.5 Principal ideal domains | p. 167 |
7.6 Irreducible polynomials | p. 170 |
7.7 Lagrange interpolation | p. 173 |
7.8 Fields of fractions | p. 175 |
7.9 Exercises | p. 178 |
7.10 Study projects | p. 182 |
7.11 Notes | p. 184 |
8 Factorization | p. 185 |
8.1 Factorization in integral domains | p. 185 |
8.2 Noetherian domains | p. 188 |
8.3 Unique factorization domains | p. 190 |
8.4 Roots of polynomials | p. 193 |
8.5 Splitting fields | p. 196 |
8.6 Uniqueness of splitting fields | p. 198 |
8.7 Structure of finite fields | p. 202 |
8.8 Galois fields | p. 204 |
8.9 Exercises | p. 206 |
8.10 Study projects | p. 210 |
8.11 Notes | p. 213 |
9 Modules | p. 215 |
9.1 Endomorphisms | p. 215 |
9.2 Representing a ring | p. 219 |
9.3 Modules | p. 220 |
9.4 Submodules | p. 223 |
9.5 Direct sums | p. 227 |
9.6 Free modules | p. 231 |
9.7 Vector spaces | p. 235 |
9.8 Abelian groups | p. 240 |
9.9 Exercises | p. 243 |
9.10 Study projects | p. 248 |
9.11 Notes | p. 251 |
10 Group Actions | p. 253 |
10.1 Actions | p. 253 |
10.2 Orbits | p. 256 |
10.3 Transitive actions | p. 258 |
10.4 Fixed points | p. 262 |
10.5 Faithful actions | p. 265 |
10.6 Cores | p. 267 |
10.7 Alternating groups | p. 270 |
10.8 Sylow Theorems | p. 273 |
10.9 Exercises | p. 277 |
10.10 Study projects | p. 283 |
10.11 Notes | p. 286 |
11 Quasigroups | p. 287 |
11.1 Quasigroups | p. 287 |
11.2 Latin squares | p. 289 |
11.3 Division | p. 293 |
11.4 Quasigroup homomorphisms | p. 297 |
11.5 Quasigroup homotopies | p. 301 |
11.6 Principal isotopy | p. 304 |
11.7 Loops | p. 306 |
11.8 Exercises | p. 311 |
11.9 Study projects | p. 315 |
11.10 Notes | p. 318 |
Index | p. 319 |