
Chapter 19
Finite Fields
Fields that have finitely many elements are important in many branches of mathe-
matics, including number theory, group theory, and projective geometry. They also
have practical applications, especially to the coding of digital communications, see
Lidl and Niederreiter (1986), and, especially for the history, Thompson (1983).
The most familiar examples of such fields are the fields Z
p
for prime p, but these
are not all. In this chapter we give a complete classification of all finite fields. It turns
out that a finite field is uniquely determined up to isomorphism by the number of
elements that it contains, that this number must be a