# prime residue class

Let $m$ be a positive integer. There are $m$ residue classes^{} $a+m\mathbb{Z}$ modulo $m$. Such of them which have

$$\mathrm{gcd}(a,m)=\mathrm{\hspace{0.33em}1},$$ |

are called the prime residue classes or prime classes modulo $m$, and they form an Abelian group^{} with respect to the multiplication^{}

$$(a+m\mathbb{Z})\cdot (b+m\mathbb{Z}):=ab+m\mathbb{Z}.$$ |

This group is called the residue class group modulo $m$. Its order is $\phi (m)$, where $\phi $ means Euler’s totient function. For example, the prime classes modulo 8 (i.e. $1+8\mathbb{Z}$, $3+8\mathbb{Z}$, $5+8\mathbb{Z}$, $7+8\mathbb{Z}$) form a group isomorphic^{} to the Klein 4-group.

The prime classes are the units of the residue class ring $\mathbb{Z}/m\mathbb{Z}={\mathbb{Z}}_{m}$ consisting of all residue classes modulo $m$.

Analogically, in the ring $R$ of integers (http://planetmath.org/ExamplesOfRingOfIntegersOfANumberField) of any algebraic number field, there are the residue classes and the prime residue classes modulo an ideal $\U0001d51e$ of $R$. The number of all residue classes is $\text{N}(\U0001d51e)$ and the number of the prime classes is also denoted by $\phi (\U0001d51e)$. It may be proved that

$$\phi (\U0001d51e)=\text{N}(\U0001d51e)\prod _{\U0001d52d|\U0001d51e}\left(1-\frac{1}{\text{N}(\U0001d52d)}\right);$$ |

N is the absolute norm of ideal and $\U0001d52d$ runs all distinct prime ideals^{} dividing $\U0001d51e$ (cf. the first formula^{} in the entry “Euler phi function (http://planetmath.org/EulerPhiFunction)”). Moreover, one has the result

$${\alpha}^{\phi (\U0001d51e)}\equiv \mathrm{\hspace{0.33em}1}\phantom{\rule{veryverythickmathspace}{0ex}}(mod\U0001d51e)$$ |

for $((a),\U0001d51e)=(1)$, 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 |