A field is a set of integers, such that operations on the integers result in other elements of the set.
Aka, A field is like a group but it has both the addition and multiplication operations.
Galois showed that for a field to be finite, the number of elements should be some power of a prime, aka $p^n$.
A Galois Field $GF(p^n)$ is a finite field with $p^n$ elements.
When $n=1$:
\[GF(p^n) = GF(p) = Z_p = \{0,1,...,p-1\}\]There is a common field, $GF(2) = {0,1}$. It has two operations, + and *.
Multiplication:
| X | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
Addition:
| + | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Inverses:
| a | -a | $a^{-1}$ |
|---|---|---|
| 0 | 1 | - |
| 1 | 0 | 1 |
When $GF(p^m)$