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  "body": "\n## Example\n\nLet $q$ be a prime power and let $\\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements.\nThe plane $\\mathrm{AG}(2, q)$ has as points the pairs $(x, y) \\in \\F_q^2$.\nIts lines are the sets $\\set{(x, y) : ax + by = c}$ with $a, b, c \\in \\F_q$ and $(a, b) \\ne (0, 0)$.\n\nCheck that it is an [[affine plane|EC7XT2ZM5S]].\nTwo distinct points $(x_1, y_1)$ and $(x_2, y_2)$ lie on exactly one line.\nIndeed, the two linear equations in $(a, b, c)$ that say both points lie on the line have a one-dimensional solution space.\nLet $ax + by = c$ be a line and $(x_0, y_0)$ a point not on it.\nThe line $ax + by = a x_0 + b y_0$ passes through the point and is disjoint from the given line.\nEvery other line through the point has a different direction, so it meets the given line.\nThe points $(0, 0)$, $(1, 0)$ and $(0, 1)$ are not on a common line.\n\nIt has $q^2$ points and $q^2 + q$ lines, each line has $q$ points, and the lines fall into $q + 1$ parallel classes.\nFor $q = 2$ it has $4$ points and $6$ lines, and every pair of points is a line.\nAdding one point at infinity for each parallel class and a line at infinity gives the [[projective plane $\\mathrm{PG}(2, q)$|CSMB49BDPT]].\n\n## Explanations\n\n### L1 human\n\nTake the usual coordinate plane, but let the coordinates run over a finite field instead of the real numbers.\nLines are still the solution sets of equations $ax + by = c$, and two points still determine a line.\nParallel lines still exist and never meet.\nThis gives finite planes with $q^2$ points, one for each prime power $q$.\n",
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