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# 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

$\gcd(a,\,m)\;=\;1,$ |

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 $\varphi(m)$, where $\varphi$ 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 of any algebraic number field, there are the residue classes and the prime residue classes modulo an ideal $\mathfrak{a}$ of $R$. The number of all residue classes is $\mbox{N}(\mathfrak{a})$ and the number of the prime classes is also denoted by $\varphi(\mathfrak{a})$. It may be proved that

$\varphi(\mathfrak{a})\;=\;\mbox{N}(\mathfrak{a})\prod_{{\mathfrak{p}|\mathfrak% {a}}}\left(1-\frac{1}{\mbox{N}(\mathfrak{p})}\right);$ |

N is the absolute norm of ideal and $\mathfrak{p}$ runs all distinct prime ideals dividing $\mathfrak{a}$ (cf. the first formula in the entry “Euler phi function”). Moreover, one has the result

$\alpha^{{\varphi(\mathfrak{a})}}\;\equiv\;1\;\;(\mathop{{\rm mod}}\mathfrak{a})$ |

for $((a),\,\mathfrak{a})=(1)$, generalising the Euler–Fermat theorem.

## Mathematics Subject Classification

20K01*no label found*13M99

*no label found*11A07

*no label found*

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