Algebra : A Computational Introduction

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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(R) 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 more

Product details

  • Hardback | 336 pages
  • 159.5 x 240.5 x 21.6mm | 632.01g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • New.
  • 6 black & white tables, 16 black & white halftones
  • 1584880643
  • 9781584880646

Review quote

"emphasizes the computational aspects of modern abstract algebraauthor has integrated the software Mathematica into the discussions-especially in the group theory sections-but is careful not to make any logical reliance on this software. For one wishing to see the theory unfold through a highly computational approach, this text has much to recommend writing is logical but not excessively formalI feel that this text was very courageously written[the] focus is a bit more narrow that that of the typical first-year undergraduate course in abstract algebra. Yet, if one wishes to develop a deep and intuitive rapport with basic group and Galois theory, then this text has much to offer." --David B. Surowski, in Mathematical Reviews, Issue 2001ishow more

Table of contents

CONGRUENCES Basic Properties Divisibility Tests Common Divisors Solving Congruences The Integers Modulo n Introduction to Software PERMUTATIONS Permutations as Mappings Cycles Sign of a Permutation PERMUTATION GROUPS Definition Cyclic Groups Generators Software and Calculations LINEAR GROUPS Definitions and Examples Generators Software and Calculations GROUPS Basic Properties and More Examples Homomorphisms SUBGROUPS Definition Orthogonal Groups Cyclic Subgroups and Generators Kernel and Image of a Homomorphism SYMMETRY GROUPS Symmetries of Regular Polygons Symmetries of Platonic Solids Improper Symmetries Symmetries of Equations GROUP ACTIONS Examples Orbits and Stabilizers Fractional Linear Transformations Cayley's Theorem Software and Calculations COUNTING FORMULAS The Class Equation A First Application Burnside's Counting Lemma Finite Subgroups of SO(3) COSETS Lagrange's Theorem Normal Subgroups Quotient Groups The Canonical Isomorphism Software and Calculations SYLOW SUBGROUPS The Sylow Theorems Groups of Small Order A List A Calculation SIMPLE GROUPS Composition Series Simplicity of An Simplicity of PSL(2,Fp) ABELIAN GROUPS Free Abelian Groups Row and Column Reduction of Integer Matrices Classification Theorems Invariance of Elementary Divisors The Multiplicative Group of the Integers Mod n POLYNOMIAL RINGS Basic Properties of Polynomials Unique Factorization into Irreducibles Finding Irreducible Polynomials Commutative Rings Congruences Factoring Polynomials over a Finite Field Calculations SYMMETRIC POLYNOMIALS Polynomials in Several Variables Symmetric Polynomials and Functions Sums of Powers Discriminants Software ROOTS OF EQUATIONS Introduction Extension Fields Degree of an Extension Splitting Fields Cubics Cyclotomic Polynomials Finite Fields Plots and Calculations GALOIS GROUPS Introduction Definition How Large is the Galois Group? The Galois Correspondence Discriminants QUARTICS Galois Groups of Quartics The Geometry of the Cubic Resolvent Software THE GENERAL EQUATION OF THE nth DEGREE Examples Symmetric Functions The Fundamental Theorem of Algebra SOLUTION BY RADICALS Formulas for a Cubic Cyclic Extensions Solution by Radicals in Higher Degrees Calculations RULER-AND-COMPASS CONSTRUCTIONS Introduction Algebraic Interpretation Construction of Regular Polygons Periods APPENDIX: MATHEMATICA COMMANDSshow more

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