Zero divisor
Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.
Numbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
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.
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.
A number that is its own square
Square 625 and the answer, 390625, ends in 625. Square 9376 and it ends in 9376. Keep extending them and two endless strings of digits appear, running leftward, each equal to its own square — and their product is nought, though neither is. Base ten does this because ten is two times five: a number in base ten is secretly a pair of numbers, one in base two and one in base five, and these two strings are the pair (one, nought) and (nought, one). In a prime base no such thing can happen.
Named alongside it
The objects these essays reach for when they reach for this one.
Modular arithmeticFinite fieldPrime powerBijectionCharacteristicChinese remainder theoremClosureCompositeConstructionCounterexampleCounting argumentCyclic group