# A Concrete Introduction to Higher Algebra

• Lindsay N. Childs
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xiv
2. ### Numbers

1. Pages 3-7
2. Pages 9-25
3. Pages 27-52
4. Pages 53-70
5. Pages 71-89
3. ### Congruence classes and rings

1. Pages 93-121
2. Pages 123-146
3. Pages 147-167
4. ### Congruences and Groups

1. Pages 171-200
2. Pages 201-221
3. Pages 223-252
4. Pages 253-281
5. ### Polynomials

1. Pages 285-293
2. Pages 295-306
3. Pages 307-338
4. Pages 339-353
5. Pages 355-371
6. Pages 373-383
6. ### Primitive Roots

1. Pages 387-412
2. Pages 413-431
3. Pages 433-457
4. Pages 459-475
7. ### Finite Fields

1. Pages 479-493
2. Pages 495-510
3. Pages 511-527
8. ### Factoring Polynomials

1. Pages 531-556
2. Pages 557-567
9. Back Matter
Pages 569-603

### Introduction

This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. A strong emphasis on congruence classes leads in a natural way to finite groups and finite fields. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, error correction, integration, and especially to elementary and computational number theory. The later chapters include expositions of Rabin's probabilistic primality test, quadratic reciprocity, the classification of finite fields, and factoring polynomials over the integers. Over 1000 exercises, ranging from routine examples to extensions of theory, are found throughout the book; hints and answers for many of them are included in an appendix.

The new edition includes topics such as Luhn's formula, Karatsuba multiplication, quotient groups and homomorphisms, Blum-Blum-Shub pseudorandom numbers, root bounds for polynomials, Montgomery multiplication, and more.

"At every stage, a wide variety of applications is presented...The user-friendly exposition is appropriate for the intended audience"

- T.W. Hungerford, Mathematical Reviews

"The style is leisurely and informal, a guided tour through the foothills, the guide unable to resist numerous side paths and return visits to favorite spots..."

- Michael Rosen, American Mathematical Monthly

### Keywords

algebra field finite group homomorphism matrices number theory

### Editors and affiliations

• Lindsay N. Childs
• 1
1. 1.Department of MathematicsUniversity at Albany State University of New YorkAlbanyUSA