Projective line over a finite field
This page contains constructions of the projective line over all finite fields F_{q} up to F_{7}, including esoteric F_{1}. Any F_{q}P^{1} consists of q+1 points.
Images show elements of the Cartesian square F^{2} of the field as colored discs, where the same color means proportionality, i.e. representing the same point of the projective line. Except for F_{4}, 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).
F_{3}, F_{5} and F_{7}[edit]
These fields have odd number of elements. They are arranged symmetrically with 0 in the center.
F_{3} = {−1=2, 0, 1} | . . . . | F_{5} = {−2=3, −1=4, 0, 1, 2} |
color | coordinates | equation | . . | color | coordinates | equation |
---|---|---|---|---|---|---|
| | [0:1] | x = 0 | (F_{5} only) | |||
— | [1:0] | y = 0 | × | [1:2] | y = 2x | |
／ | [1:1] | y = x | × | [1:−2] | y = −2x | |
＼ | [1:−1] | y = −x |
F_{7} = {−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 |
F_{2} and F_{1}[edit]
In F_{2} 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 F_{1} 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 F_{1}, hence only {x = 0} ∪ {y = 0} is shown.
F_{2} = {0, 1} (0 is center) | . . . . | F_{1} (not a set) |
color | coordinates | equation | . . | color | description |
---|---|---|---|---|---|
| | [0:1] | x = 0 | ■ | undefined (F_{1} only) | |
— | [1:0] | y = 0 | |||
× | [1:1] | y = x (F_{2} only) |
F_{4}[edit]
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 F_{4}, because F_{4} is not a quotient of the ring of integers. This picture uses a complex representation of F_{4} seen as {0, 1, α, α²}, where – a cubic root of unity. For consistent addition we must set .
F_{4}^{2} contains 15 non-zero elements, to each of them corresponds a 2-dimensional complex vector. The real projectivisation of C^{2} is RP^{3}. In this representation any 1-dimensional subspaces of F_{4}^{2} is a line in RP^{3}, which contains exactly 3 points of F_{4}^{2}\{0}.
F_{4}^{2} is a 4-dimensional linear space over F_{2}. These subspaces are also 2-dimensional F_{2}-subspaces (or, the same, F_{2}-lines on F_{2}P^{3}). Some other lines of 3 points, which are visible on the picture, are F_{2}-lines, but not all F_{2}-lines are shown.
With an appropriate choice of affine part of the RP^{3}, 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 RP^{3}.
Another approach to visualization of F_{4}^{2} is the golden ratio and geometrical structures with rotational symmetry of order 5.
- Let:
Then we have . If we suppose addition modulo 2, then , exactly what we must have for F_{4}. 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[edit]
These images of F_{p}P^{1} for all prime p (p=2,3,5,7) fit for a rectangular tiling.