| Description |
1 online resource (x, 319 pages) : illustrations |
| Series |
Studies in advanced mathematics |
|
Studies in advanced mathematics.
|
| Contents |
1 Congruences 1 -- Basic Properties 1 -- Divisibility Tests 2 -- Common Divisors 6 -- Solving Congruences 9 -- The Integers Modulo n 11 -- Introduction to Software 13 -- 2 Permutations 19 -- Permutations as Mappings 19 -- Cycles 20 -- Sign of a Permutation 23 -- 3 Permutation Groups 27 -- Cyclic Groups 29 -- Generators 30 -- Software and Calculations 32 -- 4 Linear Groups 39 -- Generators 41 -- Software and Calculations 45 -- 5 Groups 51 -- Basic Properties and More Examples 51 -- Homomorphisms 56 -- 6 Subgroups 63 -- Orthogonal Groups 63 -- Cyclic Subgroups and Generators 65 -- Kernel and Image of a Homomorphism 70 -- 7 Symmetry Groups 75 -- Symmetries of Regular Polygons 76 -- Symmetries of Platonic Solids 78 -- Improper Symmetries 83 -- Symmetries of Equations 84 -- 8 Group Actions 89 -- Orbits and Stabilizers 90 -- Fractional Linear Transformations 94 -- Cayley's Theorem 97 -- Software and Calculations 98 -- 9 Counting Formulas 105 -- The Class Equation 105 -- A First Application 110 -- Burnside's Counting Lemma 110 -- Finite Subgroups of SO(3) 113 -- 10 Cosets 119 -- Lagrange's Theorem 119 -- Normal Subgroups 123 -- Quotient Groups 125 -- The Canonical Isomorphism 126 -- Software and Calculations 129 -- 11 Sylow Subgroups 137 -- The Sylow Theorems 137 -- Groups of Small Order 141 -- A List 145 -- A Calculation 146 -- 12 Simple Groups 149 -- Composition Series 149 -- Simplicity of A[subscript n] 151 -- Simplicity of P S L (2, F[subscript p]) 153 -- 13 Abelian Groups 159 -- Free Abelian Groups 159 -- Row and Column Reduction of Integer Matrices 162 -- Classification Theorems 166 -- Invariance of Elementary Divisors 169 -- The Multiplicative Group of the Integers Mod n 172 -- 14 Polynomial Rings 177 -- Basic Properties of Polynomials 178 -- Unique Factorization into Irreducibles 183 -- Finding Irreducible Polynomials 185 -- Commutative Rings 189 -- Congruences 192 -- Factoring Polynomials over a Finite Field 197 -- Calculations 201 -- 15 Symmetric Polynomials 209 -- Polynomials in Several Variables 209 -- Symmetric Polynomials and Functions 210 -- Sums of Powers 214 -- Discriminants 215 -- Software 216 -- 16 Roots of Equations 221 -- Extension Fields 223 -- Degree of an Extension 225 -- Splitting Fields 228 -- Cubics 231 -- Cyclotomic Polynomials 233 -- Finite Fields 236 -- Plots and Calculations 238 -- 17 Galois Groups 245 -- How Large is the Galois Group? 250 -- The Galois Correspondence 254 -- Discriminants 265 -- 18 Quartics 269 -- Galois Groups of Quartics 269 -- The Geometry of the Cubic Resolvent 272 -- Software 275 -- 19 The General Equation of the nth Degree 279 -- Symmetric Functions 280 -- The Fundamental Theorem of Algebra 282 -- 20 Solution by Radicals 285 -- Formulas for a Cubic 285 -- Cyclic Extensions 288 -- Solution by Radicals in Higher Degrees 290 -- Calculations 294 -- 21 Ruler-and-Compass Constructions 297 -- Algebraic Interpretation 298 -- Construction of Regular Polygons 302 -- Periods 304 -- Appendix Mathematica Commands 309 |
| Summary |
"In Algebra: A Computational Introduction, the author demonstrates how software can be used as a problem-solving tool for algebra." "The author introduces permutation groups, then linear groups, before tackling abstract groups. He 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 prepare readers for understanding the more abstract concepts." "By integrating the use of Mathematica throughout the book in examples and exercises, the author helps readers establish an understanding and appreciation of the material. The exercises and examples along with downloads available from the Internet help establish a working knowledge of Mathematica and provide a reference for complex problems encountered in the field."--Jacket |
| Bibliography |
Includes bibliographical references (page 313) and index |
| Notes |
Print version record |
| Subject |
Algebra.
|
|
algebra.
|
|
MATHEMATICS -- Algebra -- Intermediate.
|
|
Algebra
|
|
Computeralgebra
|
|
Algebra.
|
|
Mathematica (computerprogramma)
|
|
Matemática da computação.
|
|
Álgebra computacional.
|
|
ALGEBRA.
|
|
NONLINEAR EQUATIONS.
|
|
LINEAR EQUATIONS.
|
|
ANALYSIS (MATHEMATICS)
|
| Form |
Electronic book
|
| ISBN |
9781482295696 |
|
1482295695 |
|
9781315273167 |
|
1315273160 |
|
9781351992534 |
|
1351992538 |
|
