Part III · Finite fields · Lesson 7

Fields with pᵏ elements, built from polynomials

Polynomials over GF(p) modulo an irreducible polynomial, where irreducibles play the role of primes.

This lesson is planned. Here is what it will cover.

You’ll learn

• Elements are remainders $a_0 + a_1 x + \dots + a_{k-1} x^{k-1}$ modulo a polynomial $f$ of degree $k$.
• Modulo a reducible $f$, zero divisors appear just as in $\mathbb{Z}/6\mathbb{Z}$; modulo an irreducible $f$, every nonzero element is invertible.
• Building $\mathrm{GF}(4)$, $\mathrm{GF}(8)$, and $\mathrm{GF}(9)$ by hand, and why $\mathrm{GF}(4) \ne \mathbb{Z}/4\mathbb{Z}$.