Finite field
Named by 9 essays across one field — each of them below, with the objects they name alongside it.
A polynomial through the gaps
Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.
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.
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.
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.
The thirty-six officers
Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.
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.
The plane hiding in the squares
A complete family of orthogonal squares is not a collection of squares that happen to agree nowhere. It is a geometry — a plane with n² points in which every two points lie on exactly one line — and reading it that way is how the impossible orders were found.
A memory of four bits
A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.
Nineteen thousand bits of state
The generator most simulations actually use is not clever. It is a linear recurrence over the two-element field with an enormous state, and its virtues are a proved period, a proved equidistribution and speed — none of which is unpredictability, which it does not have and does not claim.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting argumentProjective planeLatin squareModular arithmeticOrthogonal latin squaresPolynomialPrime powerCounterexampleDiscrete logarithmFano planeIncidencePrimitive element