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.