Part III · Finite fields · Lesson 8

One element to generate them all

Primitive elements, log tables, and a fast test for primitive polynomials.

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

You’ll learn

• A primitive element $\alpha$ has powers that visit every nonzero element, so multiplication becomes adding exponents.
• Log tables for small fields.
• Testing primitivity without trying every power: $\alpha^{(p^k - 1)/r} \ne 1$ for each prime $r \mid p^k - 1$.