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Concept

Prime power

A prime raised to a whole-number exponent, which is exactly the sizes at which a finite field exists. A finite field exists at exactly these sizes and no others, which is why there is no field with six elements.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

computation · Finite fields The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Seven points, seven lines

A geometry with seven points, in which every two points lie on exactly one line and every two lines meet in exactly one point. There are no parallels, the whole thing is built out of the two-element field, and one of its lines has to be drawn as a circle.

computation · Finite geometry The 3 mutually orthogonal squares of order 4. Every Latin square built from the field of order 4 as a·i + j, one for each non-zero multiplier, with every pair checked orthogonal.

A field's worth of squares

Two orthogonal squares of order five are easy to stumble on. Four of them, every pair orthogonal, is not a stumble — it is one line of arithmetic over a field, and the field supplies as many as the order allows.

computation · Latin squares A plane of 13 points from a list of 4 numbers. A ring of 13 points with one block of 4 of them drawn as a closed path, beside the table of the 13 blocks its shifts produce.

A plane in a list of numbers

A projective plane of order three has thirteen points and thirteen lines and fifty-two incidences. All of it is in the four numbers 0, 1, 3, 9 — because their pairwise differences hit every non-zero residue modulo thirteen exactly once, and the plane is that list's thirteen shifts.

computation · Finite geometry The powers of 2 in 1 to 12: one number, 8, stands alone. The whole numbers from 1 to 12, each with a bar whose height is the power of 2 dividing it. A single number has the tallest bar, which is why the harmonic number H(12) has an even denominator and an odd numerator.

The sum that steps over every whole number

The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.

analysis · Harmonic series The plane of order 3 as a table, and the table times its transpose. The 13 × 13 incidence table of the projective plane of order 3 and its product with its transpose, which has 4 on the diagonal and 1 in every other cell.

The orders a plane cannot have

Every counting condition allows a projective plane of order six, and there is none. The proof that rules it out looks at one matrix identity — each point on seven lines, each two points on one — and turns it, by way of Lagrange's four squares, into the statement that six would have to be a sum of two squares. Run on the planes that do exist, the same argument hands back their orders as sums of two squares; run on six, it asks for something no arithmetic can supply.

computation · Finite geometry

Named alongside it

The objects these essays reach for when they reach for this one.

Finite fieldProjective planeCounting argumentIncidenceModular arithmeticZero divisorBasisBijectionCharacteristicClosureConstructionCounterexample

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