Part IV · Planes from fields · Lesson 13

The Singer cycle

Multiplying by α rotates the whole plane, so PG(2, q) is cyclic.

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

You’ll learn

• Multiplication by $\alpha$ is $\mathrm{GF}(q)$-linear, so it sends lines to lines.
• It moves point $i$ to point $i + 1$, cycling through all $v$ points.
• So any one line of $\mathrm{PG}(2, q)$ is a $(q^2 + q + 1, q + 1, 1)$ difference set.