Concept

Zero divisor

A non-zero element that multiplies with another non-zero element to give zero, which no field can contain. Their absence is what makes cancellation legal, and a dial has none exactly when its size is prime.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

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.

discrete · Modular arithmetic

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 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

Named alongside it

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

Modular arithmetic Finite field Prime power Bijection Characteristic Closure Composite Construction Counterexample Counting argument Cyclic group Fermats little theorem

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