If the modulus g(x) is an irreducible polynomial of degree m over GF(p), then the finite field GF(pm) can be constructed by the set of polynomials over GF(p) whose degree is at most m-1, where addition and multiplication are done modulo g(x).
For example, the finite field GF(32) can be constructed as the set of polynomials whose degrees are at most 1, with addition and multiplication done modulo the irreducible polynomial x2+1 (you can also choose another modulus, as long as it is irreducible and has degree 2).
Table 4.1. Addition modulo x2+1
+ | 0 | 1 | 2 | x | x+1 | x+2 | 2x | 2x+1 | 2x+2 |
---|---|---|---|---|---|---|---|---|---|
0 | 0 | 1 | 2 | x | x+1 | x+2 | 2x | 2x+1 | 2x+2 |
1 | 1 | 2 | 0 | x+1 | x+2 | x | 2x+1 | 2x+2 | 2x |
2 | 2 | 0 | 1 | x+2 | x | x+1 | 2x+2 | 2x | 2x+1 |
x | x | x+1 | x+2 | 2x | 2x+1 | 2x+2 | 0 | 1 | 2 |
x+1 | x+1 | x+2 | x | 2x+1 | 2x+2 | 2x | 1 | 2 | 0 |
x+2 | x+2 | x | x+1 | 2x+2 | 2x | 2x+1 | 2 | 0 | 1 |
2x | 2x | 2x+1 | 2x+2 | 0 | 1 | 2 | x | x+1 | x+2 |
2x+1 | 2x+1 | 2x+2 | 2x | 1 | 2 | 0 | x | x+2 | x+2 |
2x+2 | 2x+2 | 2x | 2x+1 | 2 | 0 | 1 | x+2 | x | x+1 |
Table 4.2. Multiplication modulo x2+1
+ | 0 | 1 | 2 | x | x+1 | x+2 | 2x | 2x+1 | 2x+2 |
---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | x | x+1 | x+2 | 2x | 2x+1 | 2x+2 |
2 | 0 | 2 | 1 | 2x | 2x+2 | 2x+1 | x | x+2 | x+1 |
x | 0 | x | 2x | 2 | x+2 | 2x+2 | 1 | x+1 | 2x+1 |
x+1 | 0 | x+1 | 2x+2 | x+2 | 2x | 1 | 2x+1 | 2 | x |
x+2 | 0 | x+2 | 2x+1 | 2x+2 | 1 | x | x+1 | 2x | 2 |
2x | 0 | 2x | x | 1 | 2x+1 | x+1 | 2 | 2x+2 | x+2 |
2x+1 | 0 | 2x+1 | x+2 | x+1 | 2 | 2x | 2x+2 | x | 1 |
2x+2 | 0 | 2x+2 | x+1 | 2x+1 | x | 2 | x+2 | 1 | 2x |