Finite fields — the series

The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

part 1 · computation

The non-zero elements of GF(16) as the powers of one of them. A ring of the field's non-zero elements in the order the powers of a primitive element produce them, beside a table of exponents.

Every element is a power of one of them

Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.

part 2 · computation

The zeros of x² + y² + z² over GF(5), and of x² + y² over GF(7). Grids of every point over a small prime field with the solutions of a quadratic equation filled in: the three-variable equation drawn as one slice per value of z, beside a two-variable equation with far fewer solutions.

Solutions that come in multiples of p

Count the solutions of x² + y² + z² = 0 in the field with five elements and there are 25; with seven, there are 49. Whenever a system of equations has more unknowns than its total degree, its number of solutions is a multiple of the characteristic — which forces a solution besides zero, and the reason is a sum over the field that vanishes because its non-zero elements form one cycle.

part 3 · computation

The cubic curves over GF(43) with the most and the fewest points. The solutions of two equations y squared equals x cubed plus ax plus b over the field with 43 elements, drawn as dots on a square grid: the curve with the most points and the curve with the fewest.

Give or take twice the square root

A cubic curve over the integers mod 43 should have about 44 points — one for each value of x, on average, and one at infinity. No curve misses by more than 13, the largest whole number below 2√43, and every count from 31 to 57 belongs to some curve. The first fact is Hasse's theorem, the second Deuring's, and the way the counts spread between the limits is a semicircle.

part 4 · computation

A line in every direction in the plane over GF(7), in 31 points. A square grid of the points of a small finite plane with the points of a Kakeya set filled, beside a list of the lines it contains, one for each direction.

No set with a line in every direction is small

In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.

part 5 · computation

A sum of two sets modulo a prime cannot be small

Add every element of one set of residues to every element of another. Over the whole numbers the sums always number at least |A| + |B| − 1. Modulo a prime the sums can wrap round and collide, and still they never number fewer — the theorem Cauchy proved in 1813 and Davenport again in 1935. Modulo 12 they can. A polynomial of low degree explains the difference in a paragraph.

part 6 · computation

Eighty-one cards and twenty with no SET among them. A three-by-three arrangement of three-by-three grids covering the eighty-one points of four-dimensional space modulo three, with twenty cells marked that contain no three on a line.

Twenty cards with no set among them

The card game SET is a four-dimensional space over the integers mod 3, and a set is a line in it. Twenty cards can avoid every line and twenty-one cannot — a fact that took a proof in 1970 — while laying cards down at random and stopping when nothing more fits reaches twenty about once in two thousand tries.

part 7 · computation

The polynomial bound falls exponentially behind the space. A log-scale plot against the dimension of the number of points of the space, the polynomial method's bound on a cap, and the known largest caps: the bound's line is less steep than the space's and pulls away from it.

The polynomial that bounds the caps

For forty years the best bound on a set of SET cards with no set among them shrank only like one over the dimension. In 2016 a two-page argument made it shrink exponentially, and the whole proof is a count of monomials: a table that is diagonal on a cap, one polynomial that describes it, and the fact that three parts of a degree cannot all be large.

part 8 · computation