prime residue class

Let m be a positive integer. There are m residue classesMathworldPlanetmathPlanetmath a+m modulo m.  Such of them which have

gcd(a,m)= 1,

are called the prime residue classes or prime classes modulo m, and they form an Abelian groupMathworldPlanetmath with respect to the multiplicationPlanetmathPlanetmath


This group is called the residue class group modulo m. Its order is φ(m), where φ means Euler’s totient function. For example, the prime classes modulo 8 (i.e. 1+8, 3+8, 5+8, 7+8) form a group isomorphicPlanetmathPlanetmathPlanetmath to the Klein 4-group.

The prime classes are the units of the residue class ring   /m=m   consisting of all residue classes modulo m.

Analogically, in the ring R of integers ( of any algebraic number field, there are the residue classes and the prime residue classes modulo an ideal 𝔞 of R. The number of all residue classes is N(𝔞) and the number of the prime classes is also denoted by φ(𝔞).  It may be proved that


N is the absolute norm of ideal and 𝔭 runs all distinct prime idealsMathworldPlanetmathPlanetmath dividing 𝔞 (cf. the first formulaMathworldPlanetmathPlanetmath in the entry “Euler phi function (”).  Moreover, one has the result

αφ(𝔞) 1(mod𝔞)

for  ((a),𝔞)=(1),  generalising the Euler–Fermat theoremMathworldPlanetmath (

Title prime residue class
Canonical name PrimeResidueClass
Date of creation 2013-03-22 15:43:12
Last modified on 2013-03-22 15:43:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Definition
Classification msc 20K01
Classification msc 13M99
Classification msc 11A07
Synonym prime class
Related topic MultiplicativeOrderOfAnIntegerModuloM
Related topic NonZeroDivisorsOfFiniteRing
Related topic GroupOfUnits
Related topic PrimitiveRoot
Related topic ResidueSystems
Related topic Klein4Group
Related topic EulerPhifunction
Related topic SummatoryFunctionOfArithmeticFunction
Defines residue class group