22 Finite Fields

This theory [of finite fields] is of considerable interest in its own right and it provides a particularly beautiful example of how the general theory of the preceding chapters fits together to provide a rather detailed description of all finite fields.

RICHARD A. DEAN, Elements of Abstract Algebra

In this, our final chapter on field theory, we take up one of the most beautiful and important areas of abstract algebra—finite fields. Finite fields were first introduced by Galois in 1830 in his proof of the unsolvability of the general quintic equation.

In the past 50 years, there have been important applications of finite fields in computer science, coding theory, information theory, and cryptography. But, besides the many uses of finite fields in pure and applied mathematics, there is yet another good reason for studying them. They are just plain fun!

Finite Fields

Def (finite field): A field (F,+,·) is called a finite field if the set F is finite.

Examle: Zp (p prime) with + and * mod p, is a finite field.

1. (Zp, +) is an abelian group(0 is identity)

2. (Zp \ 0, *) is an abelian group(1 is identity)

3. Distributiolian: a*(b+c) = a*b + a*c

4. Cancellationu: a*0 = 0

Classification of Finite Fields

Theorem 22.1 Classification of Finite Fields

For each prime p and each positive integer n, there is, up to isomorphism, a unique finite field of order pn.

Because there is only one field for each

prime-power pn, we denote it by GF(pn), in

honor of Galois, and call it the Galois field

of order pn.

Galois Fields GF(pk)

Properties of a Finite Field

It can be shown that finite fields have order pn, where p is a prime.

It can be shown that for each prime p and each positive integer n, there is, up to isomorphism, a unique finite field of order pn.

Let GF(pn) represent a finite field of order pn..

Structure of Finite Fields

 

Corollary 1

[GF(pn):GF(p)] = n

Corollary 2 GF(pn) Contains an Element of Degree n

Let a be a generator of the group of nonzero elements of GF( pn) under multiplication. Then a is algebraic over

GF( p) of degree n.

Subfields of a Finite Field

Theorem 22.3 Subfields of a Finite Field

For each divisor m of n, GF( pn) has a unique subfield of order pm. Moreover, these are the only subfields of GF( pn).

EXAMPLES Let F be the field of order 16 given. Then there

are exactly three subfields of F, and their orders are 2, 4, and 16. Obviously, the subfield of order 2 is {0, 1} and the subfield of order 16 is F itself. To find the subfield of order 4, we merely observe that the three nonzero elements of this subfield must be the cyclic subgroup of F* = <x> of order 3. So the subfield of order 4 is

{0, 1, x5, x10} = {0, 1, x2 + x, x2 + x + 1}.

PREPARED BY:

TONY ARCELLANA CERVERA JR.

III-BSMATH - PURE