TABLE OF CONTENTS In this section, we show the construction for the finite field GF( p n). In Section 3.1, we have already seen the finite field GF( p) = 
p of order p where p is a prime. The elements of GF( p) are {0 , 1 , , p 1}, and the addition + and and multiplication are carried out modulo p. We need the following fact (see van der Waerden (1953)) to construct the finite field GF( p n).
For every prime p and every degree n > 1 , there is at least one irreducible polynomial of degree n over p .
Let n be a positive integer. To construct the finite field GF( p n) of order p n, we choose f( x) to be an irreducible polynomial over GF( p) of degree n. Let us agree that ? is a formal symbol that satisfies f( ?) = 0. Let
We define two operations: + and on GF( p n) as follows. For g( ?) , h( ?) ? GF( p n),
| Addition: | |
| Multiplication: | g( ?) h( ?) = r( ?) |
where r( ?) is computed as follows:
Multiply g(
