Algebra: A Computational Introduction

ISBN-13: 9781584880646 / Angielski / Twarda / 2000 / 336 str.

John Scherk

Opis

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 field.

Recenzje

"… emphasizes the computational aspects of modern abstract algebra…author 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 formal…I 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 2001i

Spis treści

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 COMMANDS

O autorze

John Scherk