TABLE OF CONTENTS Fermat s theorem (Corollary 3.2) implies that every element ? of GF( q) where q is a prime or a power of a prime, say q = p n, satisfies the equation
This polynomial has all its coefficients from the prime field GF( p) and is monic. However, ? may satisfy an equation with a lower degree than in Eq. (3.1).
Let ? be an element in GF( p n) . The minimal polynomial of ? over GF( p) is defined as the lowest degree monic polynomial m( x) ? GF( p)[ x] such that m( ?) = 0 .
Note. The minimal polynomial of any element in GF( p n) is unique.
In GF(2 4), defined by ? 4 + ? + 1 = 0, the minimal polynomials, having coefficients equal to 0 or 1, are listed in the following table.
| Element | Minimal polynomial |
|---|---|
| 0 | x |
| 1 | x + 1 |
| ? | x 4 + x + 1 |
| ? 1 = ? 14 | x 4 + x 3 + 1 |
| ? 3 | x 4 + x 3 + x 2 + x + 1 |
| ? 5 | x 2 + x + 1 |
We will show a method for finding minimal polynomials later.
