Classification of Finite Fields

Abstract Algebra · Axiom Academy

LESSON Classification of Finite Fields The fundamental structure theorem: every finite field has prime power order, and fields of the same order are uniquely isomorphic. 1. The Order of a Finite Field The order (or cardinality) of a finite field is the number of elements it contains. This number cannot be arbitrary—it must be a prime power. The smallest finite field has 2 elements: ℤ₂ = 0, 1 Fields of order 2, 3, 4, 5, 7, 8, 9, 11, ... exist No field of order 6, 10, 12, 14, 15, ... exists (not prime powers!) The prime p is called the characteristic of the field 2. Uniqueness Up to Isomorphism For each prime power p n , there exists a unique finite field up to isomorphism. This field is denoted GF(p n ) (Galois Field) or 𝔽 p n . 3. Construction as a Splitting Field The field GF(p n ) can be constructed as the splitting field of the polynomial x p n - x over ℤ p . Start with the prime field ℤ p Consider the polynomial f(x) = x p n - x The splitting field is the smallest field containing ℤ p where f(x) splits into linear factors This splitting field has exactly p n elements: the roots of f(x) 4. Example: Constructing GF(4) Let's construct the field of 4 elements as GF(2²) using the polynomial x⁴ - x over ℤ₂. Polynomial: f(x) = x⁴ - x = x⁴ + x (in ℤ₂[x]) Factorization: x⁴ + x = x(x³ + 1) = x(x + 1)(x² + x + 1) Roots in ℤ₂: 0 and 1 are roots Extension: Need roots of x² + x + 1 (irreducible over ℤ₂) Result: GF(4) = 0, 1, α, α+1 where α² + α + 1 = 0

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