prime residue class
Let be a positive integer. There are residue classes modulo . Such of them which have
are called the prime residue classes or prime classes modulo , and they form an Abelian group with respect to the multiplication
This group is called the residue class group modulo . Its order is , where means Euler’s totient function. For example, the prime classes modulo 8 (i.e. , , , ) form a group isomorphic to the Klein 4-group.
The prime classes are the units of the residue class ring consisting of all residue classes modulo .
Analogically, in the ring of integers (http://planetmath.org/ExamplesOfRingOfIntegersOfANumberField) of any algebraic number field, there are the residue classes and the prime residue classes modulo an ideal of . The number of all residue classes is and the number of the prime classes is also denoted by . It may be proved that
N is the absolute norm of ideal and runs all distinct prime ideals dividing (cf. the first formula in the entry “Euler phi function (http://planetmath.org/EulerPhiFunction)”). Moreover, one has the result
for , generalising the Euler–Fermat theorem (http://planetmath.org/EulerFermatTheorem).
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 |