**Contemporary Mathematics**

Volume: 95;
1989;
104 pp;
Softcover

MSC: Primary 12;
Secondary 11

Print ISBN: 978-0-8218-5101-2

Product Code: CONM/95

List Price: $38.00

AMS Member Price: $30.40

MAA Member Price: $34.20

**Electronic ISBN: 978-0-8218-7683-1
Product Code: CONM/95.E**

List Price: $38.00

AMS Member Price: $30.40

MAA Member Price: $34.20

# Infinite Algebraic Extensions of Finite Fields

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*Joel V. Brawley; George E. Schnibben*

Over the last several decades there has been a renewed interest in
finite field theory, partly as a result of important applications in a number
of diverse areas such as electronic communications, coding theory,
combinatorics, designs, finite geometries, cryptography, and other portions of
discrete mathematics. In addition, a number of recent books have been devoted
to the subject. Despite the resurgence in interest, it is not widely known
that many results concerning finite fields have natural generalizations to
abritrary algebraic extensions of finite fields. The purpose of this book is
to describe these generalizations.

After an introductory chapter surveying pertinent results about finite
fields, the book describes the lattice structure of fields between the finite
field \(GF(q)\) and its algebraic closure \(\Gamma (q)\). The
authors introduce a notion, due to Steinitz, of an extended positive integer
\(N\) which includes each ordinary positive integer \(n\) as a
special case. With the aid of these Steinitz numbers, the algebraic extensions
of \(GF(q)\) are represented by symbols of the form \(GF(q^N)\).
When \(N\) is an ordinary integer \(n\), this notation agrees
with the usual notation \(GF(q^n)\) for a dimension \(n\)
extension of \(GF(q)\). The authors then show that many of the finite
field results concerning \(GF(q^n)\) are also true for
\(GF(q^N)\). One chapter is devoted to giving explicit algorithms for
computing in several of the infinite fields \(GF(q^N)\) using the
notion of an explicit basis for \(GF(q^N)\) over \(GF(q)\).
Another chapter considers polynomials and polynomial-like functions on
\(GF(q^N)\) and contains a description of several classes of permutation
polynomials, including the \(q\)-polynomials and the Dickson
polynomials. Also included is a brief chapter describing two of many potential
applications.

Aimed at the level of a beginning graduate student or advanced
undergraduate, this book could serve well as a supplementary text for a course
in finite field theory.

# Table of Contents

## Infinite Algebraic Extensions of Finite Fields

- Contents xi12 free
- Preface xiii14 free
- Chapter 1: A survey of some finite field theory 122 free
- Chapter 2: Algebraic extensions of finite fields 2142
- Chapter 3: Iterated presentations and explicit bases 3758
- Chapter 4: Polynomials and polynomial functions 6182
- 4.1. Introduction 6182
- 4.2. Irreducible Polynomials over GF(qN) 6283
- 4.3. Permutation Monomials on GF(qN) 6889
- 4.4. Permutation q-Polynomials on GF(qN) 7394
- 4.5. Dickson Polynomials over GF(qN) 79100
- 4.6. Representation of Functions of GF(qN) 81102
- 4.7. Representation of Linear Transformations on GF(qN) 84105

- Chapter 5: Two applications 87108
- Bibliography 101122