# 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 (http://planetmath.org/ExamplesOfRingOfIntegersOfANumberField) 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 (http://planetmath.org/EulerPhiFunction)”).  Moreover, one has the result

 $\alpha^{\varphi(\mathfrak{a})}\;\equiv\;1\pmod{\mathfrak{a}}$

for  $((a),\,\mathfrak{a})=(1)$,  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