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 $$\varphi(\mathfrak{a}) = \mbox{N}(\mathfrak{a})\prod_{\mathfrak{p}|\mathfrak{a}}\left(1-\frac{1}{\mbox{N}(\mathfrak{p})}\right);$$ $\mbox{N}$ 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 $$\alpha^{\varphi(\mathfrak{a})} \equiv 1 \pmod{\mathfrak{a}}$$ for $((a),\,\mathfrak{a}) = (1)$ , generalising the Euler-Fermat theorem.