Irreducible
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
One way to factor, and no other
Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.
When the label may be a matrix
A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.
A factorisation that hides its primes
Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.
Named alongside it
The objects these essays reach for when they reach for this one.
Euclid lemmaGreatest common divisorPrimesUnique factorisationCharacter tableCongruenceConjugacy classCounting two waysDirichlet theoremDivisibilityGroup representationModular arithmetic