Lecture notes in Abstract Algebra

1. 22 Finite Fields

2. 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

3.  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!

4. 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

5. 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.

6. 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.

7. Galois Fields GF(pk)

8. 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..

9. Structure of Finite Fields

10.  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.

11. 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).

12. 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}.

14. PREPARED BY:
TONY ARCELLANA CERVERA
III-BSMATH - PURE
JR.