Math 2.0

Number theory Definition

Integers modulo n

Builds on Residue classCongruence is compatible with additionCongruence is compatible with multiplication

Definition

Let nn be a positive integer. The integers modulo nn form the set ℤ/nℤ={0‾,1‾,…,n−1‾}\mathbb{Z}/n\mathbb{Z} = \{ \overline{0}, \overline{1}, \dots, \overline{n - 1} \} of the residue classes modulo nn. Addition and multiplication on ℤ/nℤ\mathbb{Z}/n\mathbb{Z} are a‾+b‾=a+b‾\overline{a} + \overline{b} = \overline{a + b} and a‾⋅b‾=ab‾\overline{a} \cdot \overline{b} = \overline{a b}.

Remarks

The operations are well defined: the result does not depend on the choice of aa and bb in their classes. This follows from compatibility with addition and compatibility with multiplication. With these operations, ℤ/nℤ\mathbb{Z}/n\mathbb{Z} is a commutative ring with nn elements, with zero 0‾\overline{0} and identity 1‾\overline{1}. Other notations are ℤn\mathbb{Z}_n and ℤ/(n)\mathbb{Z}/(n). The Lean library Mathlib writes ZMod n, and it sets ZMod 0 to be ℤ\mathbb{Z}. The ring is a field if and only if nn is prime. See integers modulo a prime form a field.

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To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionCongruence is compatible with addition, statement, no evidence levelCongruence is compati…statementCongruence is compatible with multiplication, statement, no evidence levelCongruence is compati…statementResidue class, definition, no evidence levelResidue classdefinitionIntegers modulo n, definition, no evidence levelIntegers modulo ndefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Congruence modulo n definition
  3. Congruence is compatible with addition statement
  4. Congruence is compatible with multiplication statement
  5. Residue class definition
  6. Integers modulo n definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Modular_arithmetic, Integers modulo m
About this page
Tag01N3AYAA6N
URIm2://01N3AYAA6N
Aliasesring-of-integers-modulo-n, residue-class-ring
Kinddefinition
WritingW0 stub
MSC11A07
DefinesIntegers modulo n
UsesResidue class, Congruence is compatible with addition, Congruence is compatible with multiplication

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