Adequate texts that introduce the concepts of abstract algebra are plentiful. None, however, are more suited to those needing a mathematical background for careers in engineering, computer science, the physical sciences, industry, or finance than Algebra: A Computational Introduction. Along with a unique approach and presentation, the author demonstrates how software can be used as a problem-solving tool for algebra. A variety of factors set this text apart. Its clear exposition, with each chapter building upon the previous ones, provides greater clarity for the reader. The author first introduces permutation groups, then linear groups, before finally tackling abstract groups. He carefully motivates Galois theory by introducing Galois groups as symmetry groups. He includes many computations, both as examples and as exercises. All of this works to better prepare readers for understanding the more abstract concepts.By carefully integrating the use of Mathematica® throughout the book in examples and exercises, the author helps readers develop a deeper understanding and appreciation of the material. The numerous exercises and examples along with downloads available from the Internet help establish a valuable working knowledge of Mathematica and provide a good reference for complex problems encountered in the field.
Author(s): John Scherk
Edition: 2ed.
Publisher: CRC
Year: 2009
Language: English
Pages: 419
Contents......Page 5
Preface......Page 11
Introduction to Groups......Page 15
Basic Properties......Page 17
Divisibility Tests......Page 19
Common Divisors......Page 23
Solving Congruences......Page 27
The Integers Modulo n......Page 29
Introduction to Software......Page 32
Exercises......Page 35
Permutations as Mappings......Page 39
Cycles......Page 41
Sign of a Permutation......Page 44
Exercises......Page 46
Definition......Page 49
Cyclic Groups......Page 51
Generators......Page 53
Software and Calculations......Page 56
Exercises......Page 61
Definitions and Examples......Page 65
Generators......Page 68
Software and Calculations......Page 72
Exercises......Page 76
Basic Properties and More Examples......Page 79
Homomorphisms......Page 86
Exercises......Page 91
Definition......Page 95
Orthogonal Groups......Page 96
Cyclic Subgroups and Generators......Page 98
Kernel and Image of a Homomorphism......Page 104
Exercises......Page 106
Symmetry Groups......Page 111
Symmetries of Regular Polygons......Page 112
Symmetries of Platonic Solids......Page 115
Improper Symmetries......Page 120
Symmetries of Equations......Page 121
Exercises......Page 124
Examples......Page 127
Orbits and Stabilizers......Page 129
Fractional Linear Transformations......Page 133
Cayley's Theorem......Page 137
Software and Calculations......Page 138
Exercises......Page 143
The Class Equation......Page 147
A First Application......Page 153
Burnside's Counting Lemma......Page 154
Finite Subgroups of SO(3)......Page 156
Exercises......Page 162
Lagrange's Theorem......Page 165
Normal Subgroups......Page 170
Quotient Groups......Page 173
The Canonical Isomorphism......Page 174
Software and Calculations......Page 178
Exercises......Page 185
The Sylow Theorems......Page 189
Groups of Small Order......Page 194
A List......Page 199
A Calculation......Page 200
Exercises......Page 202
Composition Series......Page 205
Simplicity of An......Page 208
Simplicity of PSL(2,Fp)......Page 210
Exercises......Page 214
Free Abelian Groups......Page 217
Row and Column Reduction of Integer Matrices......Page 221
Classification Theorems......Page 225
Invariance of Elementary Divisors......Page 229
The Multiplicative Group of the Integers Mod n......Page 232
Exercises......Page 236
Solving Equations......Page 239
Basic Properties of Polynomials......Page 241
Unique Factorization into Irreducibles......Page 248
Finding Irreducible Polynomials......Page 250
Commutative Rings......Page 255
Congruences......Page 259
Factoring Polynomials over a Finite Field......Page 266
Calculations......Page 271
Exercises......Page 275
Polynomials in Several Variables......Page 281
Symmetric Polynomials and Functions......Page 282
Sums of Powers......Page 288
Discriminants......Page 289
Software......Page 290
Exercises......Page 291
Introduction......Page 295
Extension Fields......Page 297
Degree of an Extension......Page 300
Splitting Fields......Page 304
Cubics......Page 308
Cyclotomic Polynomials......Page 310
Finite Fields......Page 314
Plots and Calculations......Page 316
Exercises......Page 320
Introduction......Page 325
Definition......Page 329
How Large is the Galois Group?......Page 332
The Galois Correspondence......Page 337
Discriminants......Page 351
Exercises......Page 353
Galois Groups of Quartics......Page 357
The Geometry of the Cubic Resolvent......Page 361
Software......Page 365
Exercises......Page 366
Examples......Page 369
Symmetric Functions......Page 371
The Fundamental Theorem of Algebra......Page 373
Exercises......Page 375
Formulas for a Cubic......Page 377
Cyclic Extensions......Page 381
Solution by Radicals in Higher Degrees......Page 384
Calculations......Page 389
Exercises......Page 390
Introduction......Page 393
Algebraic Interpretation......Page 394
Construction of Regular Polygons......Page 399
Periods......Page 401
Exercises......Page 405
Mathematica Commands......Page 407
Bibliography......Page 411
Index......Page 413
