Algebra

A Computational Introduction

John Scherk

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Print ISBN: 9780367413255
• eBook ISBN: 9781351989688
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Chapter 1

Congruences

This is an introductory chapter. The main topic is the arithmetic of congruences, sometimes called “clock arithmetic.” It leads to the construction of the integers modulo n. These are among the simplest examples of groups, as we shall see in Chapter 5. If n is a prime number, then the integers modulo n form a field. In Chapter 4, we will be looking at matrices with entries in these fields. As an application of congruences, we also discuss divisibility tests. In order to be able to solve linear congruences, we review greatest common divisors and the Euclidean algorithm.

Basic Properties

DEFINITION 1.1 Fix a natural number n. The integers a and b are congruent modulo n or mod n, written

$\begin{array}{cc}a\equiv b& \left(\text{mod}n\right),\end{array}$

if a − b is divisible by n.

For example,

$\begin{array}{ll}23\equiv 1\hfill & \left(\text{mod}\text{11}\right)\hfill \\ 23\equiv 2\hfill & \left(\text{mod}\text{7}\right)\hfill \\ 23\equiv -2\hfill & \left(\text{mod}\text{25}\right).\hfill \end{array}$

If you measure time with a 12-hour clock, then you are calculating the hour modulo 12. For example, 5 hours after 9 o’clock is not 14 o’clock but 2 o’clock. We keep track of the days by reckoning modulo 7. If today is a Wednesday, then 10 days from today will be a Saturday. January 19 was a Wednesday in the year 2000. To determine what day of the week it was in 1998, we can calculate

$\begin{array}{cc}2\cdot 365=730\equiv 2& \left(\text{mod}\text{7}\right).\end{array}$

Therefore January 19 was a Friday in 1998. Calculating modulo n is very similar to calculating in the integers. First we note that congruence modulo n is an equivalence relation.

THEOREM 1.1

(i) a ≡ a (mod n);

(ii) if a ≡ b (mod n) the...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Congruences
• 2 Permutations
• 3 Permutation Groups
• 4 Linear Groups
• 5 Groups
• 6 Subgroups
• 7 Symmetry Groups
• 8 Group Actions
• 9 Counting Formulas
• 10 Cosets
• 11 Sylow Subgroups
• 12 Simple Groups
• 13 Abelian Groups
• 14 Polynomial Rings
• 15 Symmetric Polynomials
• 16 Roots of Equations
• 17 Galois Groups
• 18 Quartics
• 19 The General Equation of the nth Degree
• 20 Solution by Radicals
• 21 Ruler-and-Compass Constructions
• Appendix: Mathematica Commands
• References
• Index