Section 7.7
Finite fields

For p prime and f an irreducible polynomial of degree n in Z/pZ[X], the quotient ring Z/pZ[X]/(f) is a field with pn elements. We shall see that any field is essentially of this form.

Let F be a finite field of order q. By a previous result, we know that q = pa, the power of a prime number p. We need another (more general) version of Fermat's little theorem.

Fermat's little theorem

For all x F,    xq = x.


We use this version of Fermat's little theorem to determine the structure of finite fields.

Characterization of finite fields

  1. Xq - X = x F (X - x).
  2. For every prime power r = pb with b | a, the subset {x F | xr = x} is a subfield of F of order r.
  3. If f Z/pZ[X] is irreducible of degree a, then F is isomorphic to Z/pZ[X]/(f).
  4. If f Z/pZ[X] is irreducible, then

    f divides Xpb - X,

    where b is the degree of f.

The last two assertions imply that if f is an irreducible polynomial in Z/pZ[X] of degree a, it factors into linear terms in F[X].