Projective line over a finite field

This page contains constructions of the projective line over all finite fields F_q up to F₇, including esoteric F₁. Any F_qP¹ 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

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}

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}

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

F₂ and F₁

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)

color	coordinates	equation	. .	color	description
|	[0:1]	x = 0		■	undefined (F1 only)
—	[1:0]	y = 0
×	[1:1]	y = x (F2 only)

F₄

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.

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 $\alpha ^{2}=\alpha ^{-1}(mod2)$, 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:

Tiling samples

These images of F_pP¹ for all prime p (p=2,3,5,7) fit for a rectangular tiling.