Part IV · Planes from fields · Lesson 12

The field hiding inside the plane

GF(q³) is GF(q)³, so every point of PG(2, q) is a power of α.

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

You’ll learn

• $\mathrm{GF}(q^3)$ is a 3-dimensional vector space over $\mathrm{GF}(q)$.
• Its nonzero elements up to $\mathrm{GF}(q)^*$ scalars are the points of $\mathrm{PG}(2, q)$.
• Because $\mathrm{GF}(q)^* = \langle \alpha^v \rangle$ with $v = q^2 + q + 1$, each point is $\alpha^i$ for exactly one $i \bmod v$.