MultiplicationTable

The multiplication table of ℤ/n, R = QuotientRing(Integers, n): the matrix whose (i, j) entry is aᵢaⱼ, its rows and columns ℤ/n's elements in the order ElementOrder lists them.

MultiplicationTable(R)the n × n matrix of products, as ResidueClass entries, for 2 ≤ n ≤ 64

MultiplicationTable(R, ElementOrder -> order)rows and columns listed Natural (0, 1, …), by their ChineseRemainder residues, or by their p-adic digits (Adic)

Typed signatures

Type	Package	Applies
(value, value*) -> list	residues	always

Domain: Modular arithmetic

Details

• The units' rows and columns form a Latin square: the multiplication table of the unit group
• Listed by Chinese remainders, the idempotents mark how n splits; listed by digits, ℤ/pᵏ's table is p × p blocks of ℤ/pᵏ⁻¹'s, each block the products of the lifts of one pair of residues
• A Show draws it at any size as ArrayPlot(MultiplicationTable(R)); as a value it stops at n = 64

Examples

Basic #

ℤ/4: 2 · 2 = 0, a nilpotent

live ↗

In$\mathrm{MultiplicationTable}(ℤ/4ℤ)$Out$\begin{pmatrix}\overline{0}_{4} & \overline{0}_{4} & \overline{0}_{4} & \overline{0}_{4}\\ \overline{0}_{4} & \overline{1}_{4} & \overline{2}_{4} & \overline{3}_{4}\\ \overline{0}_{4} & \overline{2}_{4} & \overline{0}_{4} & \overline{2}_{4}\\ \overline{0}_{4} & \overline{3}_{4} & \overline{2}_{4} & \overline{1}_{4}\end{pmatrix}$

listed 0, 4, 2, 3, 1, 5 (by residue mod 2, then mod 3): ℤ/2's table with ℤ/3's in each cell

live ↗

In$\mathrm{MultiplicationTable}(ℤ/6ℤ, \mathrm{Rule}(\mathrm{ElementOrder}, \mathrm{ChineseRemainder}))$Out$\begin{pmatrix}\overline{0}_{6} & \overline{0}_{6} & \overline{0}_{6} & \overline{0}_{6} & \overline{0}_{6} & \overline{0}_{6}\\ \overline{0}_{6} & \overline{4}_{6} & \overline{2}_{6} & \overline{0}_{6} & \overline{4}_{6} & \overline{2}_{6}\\ \overline{0}_{6} & \overline{2}_{6} & \overline{4}_{6} & \overline{0}_{6} & \overline{2}_{6} & \overline{4}_{6}\\ \overline{0}_{6} & \overline{0}_{6} & \overline{0}_{6} & \overline{3}_{6} & \overline{3}_{6} & \overline{3}_{6}\\ \overline{0}_{6} & \overline{4}_{6} & \overline{2}_{6} & \overline{3}_{6} & \overline{1}_{6} & \overline{5}_{6}\\ \overline{0}_{6} & \overline{2}_{6} & \overline{4}_{6} & \overline{3}_{6} & \overline{5}_{6} & \overline{1}_{6}\end{pmatrix}$

About

Examples	2

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no ring tables: GroupMultiplicationTable tabulates a group, and ℤ/n under multiplication is not one.

See also: AdditionTable, QuotientRing, MultiplicativeOrder