# Numbers and symmetry: an introduction to algebra

## CRC Press 1997, 260 pages

1
New Numbers
1.1
A planeful of integers, Z[i]
1.2
Circular numbers, Z_n
1.3
More integers on the number line, Z[root 2]
1.4
Notes

2
The division algorithm
2.1
Rational integers
2.2
Norms
2.2.1 Gaussian integers
2.2.2 Z[root 2]
2.3
Gaussian numbers
2.4
Q(root 2)
2.5
Polynomials
2.6
Notes

3
The Euclidean algorithm
3.1
3.2
Relatively prime numbers
3.3
Gaussian integers
3.4
Notes

4
Units
4.1
Elementary Properties
4.2
Bezout's equation
4.2.1 Casting out nines
4.3
Wilson's theorem
4.4
Orders of elements: Fermat and Euler
4.5
4.6
Z[root 2]
4.7
Notes

5
Primes
5.1
Prime numbers
5.2
Gaussian primes
5.3
Z[root 2]
5.4
Unique factorization into primes
5.5
Z_n
5.6
Notes

6
Symmetries
6.1
Symmetries of figures in the plane
6.2
Groups
6.2.1 Permutation groups
6.2.2 Dihedral groups
6.3
The cycle structure of a permutation
6.4
Cyclic groups
6.5
The alternating groups
6.5.1 Even and odd permutations
6.5.2 The sign of a permutation
6.6
Notes

7
Matrices
7.1
Symmetries and coordinates
7.2
Two-by-two matrices
7.3
The ring of matrices M_2(R)
7.4
Units
7.5
Complex numbers and quaternions
7.6
Notes

8
Groups
8.1
Abstract groups
8.2
Subgroups and cosets
8.3
Isomorphism
8.4
The group of units of a finite field
8.5
Products of groups
8.6
The Euclidean groups E(1), E(2) and E(3)
8.7
Notes

9
Wallpaper patterns
9.1
One-dimensional patterns
9.2
Plane lattices
9.3
Frieze patterns
9.4
Space groups
9.5
The 17 plane groups
9.6
Notes

10
Fields
10.1
Polynomials over a field
10.2
Kronecker's construction of simple field extensions
10.2.1 A four-element field, Kron(Z_2,X^2+X+1)
10.2.2 A sixteen-element field, Kron(Z_2,X^4+X+1)
10.3
Finite fields
10.4
Notes

11
Linear algebra
11.1
Vector spaces
11.2
Matrices
11.3
Row space and echelon form
11.4
Inverses and elementary matrices
11.5
Determinants
11.6
Notes

12
Error-correcting codes
12.1
Coding for redundancy
12.2
Linear Codes
12.2.1 A Hamming Code
12.3
Parity-check matrices
12.4
Cyclic codes
12.5
BCH codes
12.5.1 A Two-Error-Correcting Code
12.5.2 Designer Codes
12.6
CD's
12.7
Notes

13
Appendix: Induction
13.1
Formulating the n-th statement
13.2
The domino theory: iteration
13.3
Formulating the induction statement
13.3.1 Summary of steps
13.4
Squares
13.5
Templates
13.6
Recursion
13.7
Notes

14
Appendix: The usual rules
14.1
Rings
14.2
Notes

## Preface

"Suppose you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractile claws, a distinctive sonic output, etc? I'll bet not. You probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience. They should come one at a time; too many at once overload the circuits."

Ralph P. Boas, Can we make mathematics intelligible?

This book is a bridge between the plug-and-chug approach of the typical calculus course and the palace of precise crystals that is one of the great achievements of twentieth-century mathematics. It is a trip to a modern algebra zoo. We play with the animals---groups, rings, and fields---before learning their Latin names. It is a fascinating place, and you don't need a doctorate in zoology to enjoy a visit.

Number and symmetry lie at the root of modern algebra. The development of algebra can be thought of as successive extensions of the number concept, with the axioms of a ring designed to capture the essential features of a general number system. Group theory grows out of the analysis of symmetry when we go from merely counting symmetries to looking at the structure they form. Viewed in this way, a finite group is a number with structure, the two groups of order six being different avatars of the number six. Symmetry, as the study of structure preserving transformations, provides the student with a visual introduction to the central algebraic notion of an isomorphism.

The first five chapters deal mainly with the commutative rings Z, Z_n, Z[i], Z[root 2], and polynomial rings in one variable. Consideration of the groups of units of these rings leads to some elementary abelian group theory in a nontrivial concrete setting. Chapters 6 through 9 are concerned, for the most part, with groups of transformations in the plane. The idea of an abstract group, and isomorphism of groups, is introduced in Chapter 8, and these ideas are used to analyze wallpaper patterns in Chapter 9. Finite fields are constructed in Chapter 10, elementary row operations on matrices are studied in Chapter 11, and the last chapter, on coding theory, brings together this material in order to understand the mathematics of the fidelity of compact discs. There is an appendix on mathematical induction.

We have tried to get the students involved with interesting algebraic structures as quickly as possible. An encyclopedia of abstract definitions can be useful to those who only need to be reminded what the words mean, but such an encyclopedia is not suitable for beginning students, who need first to see how the words are used in practice. The point of a good definition is to organize our experience.

Mathematics textbooks are often written to conform to what is thought to be a rigorous style. Formal definitions are given before the reader has seen any of the things those definitions describe. This is the axiomatic method gone mad. Such an approach has become so pervasive that many believe that the whole point of an introductory modern algebra course is to present an abstract deductive system. To be sure, the axiomatic method is a powerful tool for understanding, but experience with concrete mathematical objects is required for it to be of any benefit. This is true even at the highest levels of research in algebra, and is of particular importance in an introductory text.

Although the students in this course must learn the rudiments of rigorous deduction---how to prove things---but they also need to learn the content of mathematics outside of an axiomatic context. Moreover, the essence of proof is that it be convincing---the truth of a theorem should be clearer after the proof than it was before. We do not prove theorems like "there is no positive integer smaller than 1." The students' first experience with proofs should not be so misleading. So even for the purpose of understanding proof, it is important to start with interesting specific mathematical structures, ask questions about them, and investigate their properties.

We hope that the readers will be sufficiently intrigued by the material to undertake some excursions on their own. We have tried to point out paths that will lead to other enjoyable experiences with algebraic structures. Certainly they will be prepared to take on any algebra textbook on the order of Topics in Algebra by Herstein. In any event, they will have experienced the basic ideas of modern algebra---the abstract study of number and symmetry.

We would like to thank Valeree Falduto for her careful and informative reading of the first half of the book, and Yuan Wang for her many helpful comments when she was teaching from the book.

## Errata

Page 5, paragraph 3: (3+5i)+(6-2i) = 9+3i, not 9-3i.

Page 5, paragraph 3: (3+5i)(6-2i) = 18+24i-10i², not 18+24i+10i². This error is repeated in paragraph 4, and in the first display on page 6.

Page 28, Exercise 1, N(rho) and N(alpha) should be |N(rho)| and |N(alpha)|.

Page 29, last line of second display: 21X³ - 17X² + 9X - 1, not 21X³ - 17X² + 5X - 1.

Page 42, just before the exercises: should be t - 1 = -n4, not t - 1 = n4, and t = 1 - 4n, not t = 4n - 1.

Page 43, first example. Should be "Try it on alpha = 4 + 3i and beta = 10 - 9i". The first table should be modified accordingly. In the second line after the first table the expression should be sigma(4 + 3i) + tau(10 - 9i).

Page 214, line 21: should be uG, not vG.