Definition. Let $\alpha$ , $\beta$ and $\kappa$ be integers of an algebraic number field $K$ and $\kappa \neq 0$ . One defines

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Theorem. The congruence ``$\equiv$ '' modulo $\kappa$ defined above is an equivalence relation in the maximal order of $K$ . There are only a finite amount of the equivalence classes, the residue classes modulo $\kappa$ .

Proof. For justifying the transitivity of ``$\equiv$ '', suppose (1) and $\beta \equiv \gamma \pmod{\kappa}$ ; then there are the integers $\lambda$ and $\mu$ of $K$ such that $\alpha\!-\!\beta = \lambda\kappa$ , $\beta\!-\!\gamma = \mu\kappa$ . Adding these equations we see that $\alpha\!-\!\gamma = (\lambda\!+\!\mu)\kappa$ with the integer $\lambda\!+\!\mu$ of $K$ . Accordingly, $\alpha \equiv \gamma \pmod{\kappa}$ .
Let $\omega$ be an arbitrary integer of $K$ and $\{\omega_1,\,\omega_2,\,\ldots,\,\omega_n\}$ a minimal basis of the field. Then we can write $$\omega = a_1\omega_1+a_2\omega_2+\ldots+a_n\omega_n,$$ where the $a_i$ 's are rational integers. For $i = 1,\,2,\,\ldots,\,n$ , the division algorithm determines the rational integers $q_i$ and $r_i$ with $$a_i = \mbox{N}(\kappa)q_i+r_i, \quad 0 \leqq r_i < |\mbox{N}(\kappa)|,$$ whence $$\omega = \mbox{N}(\kappa)(\underbrace{q_1\omega_1+q_2\omega_2+\ldots+q_n\omega_n}_{=\,\pi}) +(\underbrace{r_1\omega_1+r_2\omega_2+\ldots+r_n\omega_n}_{=\,\varrho}).$$ So we have

N

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