Constructing GF(4)

Abstract Algebra · Axiom Academy

Build a finite field with 4 elements using polynomial quotient rings Excellent work! You've successfully constructed GF(4). Here's what we learned: Finite fields from quotient rings: GF(4) = Z₂[x]/⟨x² + x + 1⟩ gives us a field with 4 elements Irreducibility is essential: We verified x² + x + 1 has no roots in Z₂, making it irreducible and ensuring the quotient is a field Four distinct elements: The cosets 0, 1, x, x+1 form all elements of GF(4) Addition is straightforward: Add polynomials coefficientwise modulo 2 Multiplication uses the relation: The key relation x² = x + 1 (from x² + x + 1 = 0) lets us reduce all products Field properties verified: Every nonzero element has a multiplicative inverse, confirming GF(4) is indeed a field This construction method generalizes! We can build GF(p^n) for any prime p and positive integer n using an irreducible polynomial of degree n over Z_p. This is fundamental in coding theory, cryptography, and computer science!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.