Projective line over a finite field

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This page contains constructions of the projective line over all finite fields Fq up to F7, including esoteric F1. Any FqP1 consists of q+1 points.

Images show elements of the Cartesian square F2 of the field as colored discs, where the same color means proportionality, i.e. representing the same point of the projective line. Except for F4, axis x is a red line from left (negative) to right (positive), and axis y is a yellow line from down (negative) to up (positive).

F3, F5 and F7[edit]

These fields have odd number of elements. They are arranged symmetrically with 0 in the center.

F3 = {−1=2, 0, 1} . . . . F5 = {−2=3, −1=4, 0, 1, 2}
Projectivisation F3P^1.svg Projectivisation F5P^1.svg
color coordinates equation . . color coordinates equation
| [0:1] x = 0 (F5 only)
[1:0] y = 0 × [1:2] y = 2x
[1:1] y = x × [1:−2] y = −2x
[1:−1] y = −x

F7 = {−3=4, −2=5, −1=6, 0, 1, 2, 3}
Projectivisation F7P^1.svg

color coordinates equation
/ [1:2] y = 2x
\ [1:−2] y = −2x
.· ́ [1:4] 2y = x
`·. [1:−4] −2y = x

F2 and F1[edit]

In F2 0 is shown at the center, and 1 is “equally distributed” between positive and negative sides, showing only a half of element at each side. In F1 only a half of (non-existent) “1” is shown at the positive side. A sum of two terms which are both non-zero is undefined over F1, hence only {x = 0} ∪ {y = 0} is shown.

F2 = {0, 1} (0 is center) . . . . F1 (not a set)
Projectivisation F2P^1.svg Projectivisation F1P^1.svg
color coordinates equation . . color description
| [0:1] x = 0 undefined (F1 only)
[1:0] y = 0
× [1:1] y = x (F2 only)

F4[edit]

Projectivisation F4P^1.svg
color coordinates equation
[0:1] x = 0
[1:0] y = 0
[1:1] y = x
[1:α] y = αx
[1:α²] αy = x







It is not easy to show the projective line over F4, because F4 is not a quotient of the ring of integers. This picture uses a complex representation of F4 seen as {0, 1, α, α²}, where \alpha = \frac{\sqrt{3}i-1}{2} – a cubic root of unity. For consistent addition we must set \alpha + 1 = \alpha^2,\ \alpha^2 + 1 = \alpha.

F42 contains 15 non-zero elements, to each of them corresponds a 2-dimensional complex vector. The real projectivisation of C2 is RP3. In this representation any 1-dimensional subspaces of F42 is a line in RP3, which contains exactly 3 points of F42\{0}.

F42 is a 4-dimensional linear space over F2. These subspaces are also 2-dimensional F2-subspaces (or, the same, F2-lines on F2P3). Some other lines of 3 points, which are visible on the picture, are F2-lines, but not all F2-lines are shown.

With an appropriate choice of affine part of the RP3, 11 points (all except (1,α), (1,α²), (α,1) and (α²,1)) lie in a tetrahedron with vertices (0,α) – yellow at the right, (α,0) – red at the left, (0,α²) – yellow at the back, (α²,0) – red at the back, obscured by (0,α) and (α²,α). The center of this tetrahedron is (1,1) – blue.

Because of complex nature of multiplication in this model, we can say that this colored lines (yellow, red, blue, cyan and magenta) show a Hopf fibration of RP3.

Divina proportio F4P^1.svg

Another approach to visualization of F42 is the golden ratio and geometrical structures with rotational symmetry of order 5.

Let: \alpha = \varphi = \frac{1+\sqrt 5}{2}

Then we have \alpha^2 = \alpha + 1,\ \alpha^{-1} = \alpha^2 - 2 = \alpha - 1. If we suppose addition modulo 2, then \alpha^2 = \alpha^{-1} (mod 2), exactly what we must have for F4. Consider two axes on the plane with angle 144° between them. If we get two vectors (2,0) and (0,2) with length 2, a lattice generated by it is shown by  (the origin is ● on this picture). The quotient space will be a torus with 15 non-zero points on it.


This is an extended and strictly periodical picture of the covering plane of that torus:

  Divina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svg
Divina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svg
  Divina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svg
Divina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svg
  Divina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svgDivina proportio F4P^1 tile.svg

Tiling samples[edit]

These images of FpP1 for all prime p (p=2,3,5,7) fit for a rectangular tiling.
Projectivisation F5P^1.svgProjectivisation F5P^1.svgProjectivisation F5P^1.svg
Projectivisation F5P^1.svgProjectivisation F5P^1.svgProjectivisation F5P^1.svg

Projectivisation F2P^1.svgProjectivisation F2P^1.svgProjectivisation F2P^1.svgProjectivisation F2P^1.svg
Projectivisation F2P^1.svgProjectivisation F2P^1.svgProjectivisation F2P^1.svgProjectivisation F2P^1.svg

See also[edit]