Affine plane over a finite field

Example

Let $q$ be a prime power and let $\F_q$ be the finite field with $q$ elements. The plane $\mathrm{AG}(2, q)$ has as points the pairs $(x, y) \in \F_q^2$. Its lines are the sets $\set{(x, y) : ax + by = c}$ with $a, b, c \in \F_q$ and $(a, b) \ne (0, 0)$.

Check that it is an affine plane. Two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ lie on exactly one line. Indeed, the two linear equations in $(a, b, c)$ that say both points lie on the line have a one-dimensional solution space. Let $ax + by = c$ be a line and $(x_0, y_0)$ a point not on it. The line $ax + by = a x_0 + b y_0$ passes through the point and is disjoint from the given line. Every other line through the point has a different direction, so it meets the given line. The points $(0, 0)$, $(1, 0)$ and $(0, 1)$ are not on a common line.

It has $q^2$ points and $q^2 + q$ lines, each line has $q$ points, and the lines fall into $q + 1$ parallel classes. For $q = 2$ it has $4$ points and $6$ lines, and every pair of points is a line. Adding one point at infinity for each parallel class and a line at infinity gives the projective plane $\mathrm{PG}(2, q)$.