F.2 EXTENSION FIELDS: EXAMPLE OF GF(2³)

When the number of field elements is not a prime but a power of a prime, we obtain an extension field. The field GF(2³) has 2³ = 8 elements, representing 3-bit symbols: 0, 1, x, x + 1, x², x² + 1, x² + x, and x² + x +1.

Arithmetic is carried out modulo a primitive polynomial of degree 3, such as p(x) = x³ + x + 1, which is irreducible over GF(2). Let α be a root of this polynomial, so that α³ = α + 1. The eight field elements can then be represented as powers of α, with α⁷ = 1.

Addition is performed by XORing corresponding coefficients, for example:

(x² + x + 1) + (x² + 1) = x (111 ⊕ 101 = 010), so α⁵ + α⁶ = α.

Multiplication is performed by multiplying polynomials and reducing modulo p(x). For example, α² × α³ = α⁵. Since α³ = α + 1, then α²(α + 1) = α³ + α² = (α + 1) + α² = α² + α + 1 = α⁵. Multiplication is cyclic: αⁱ × αʲ = α⁽ⁱ⁺ʲ⁾ (mod 7).

Extension fields such as GF(2³), GF(2⁸), and GF(2¹⁶) are fundamental in modern error-control coding schemes like BCH and Reed–Solomon codes, allowing reliable multi-bit symbol operations within a closed algebraic system.