congruence in algebraic number field

Definition.  Let $\alpha$, $\beta$ and $\kappa$ be integers (http://planetmath.org/AlgebraicInteger) of an algebraic number field $K$ and  $\kappa \ne 0$.  One defines

$\alpha \equiv \beta (mod\kappa )$(1)

if and only if  $\kappa \mid \alpha -\beta$,  i.e. iff there is an integer $\lambda$ of $K$ with $\alpha -\beta =\lambda \kappa$.

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  $\beta \equiv \gamma (mod\kappa )$; 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,  $\alpha \equiv \gamma (mod\kappa )$.
Let $\omega$ be an arbitrary integer of $K$ and  $\{{\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n\}$  a minimal basis of the field.  Then we can write

$$\omega =a_1{\omega }_1+a_2{\omega }_2+\mathrm{\dots }+a_n{\omega }_n,$$

where the $a_i$’s are rational integers.  For  $i=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },n$, the division algorithm determines the rational integers $q_i$ and $r_i$ with

whence

$$\omega =\text{N}(\kappa )(\underset{=\pi }{\underbrace{q_1{\omega }_1+q_2{\omega }_2+\mathrm{\dots }+q_n{\omega }_n}})+(\underset{=\varrho }{\underbrace{r_1{\omega }_1+r_2{\omega }_2+\mathrm{\dots }+r_n{\omega }_n}}).$$

So we have

$\omega =\text{N}(\kappa )\pi +\varrho ,$(2)

where $\pi$ and $\varrho$ are some integers of the field.  If  ${\kappa }^{(1)},{\kappa }^{(2)},\mathrm{\dots },{\kappa }^{(n)}$  are the algebraic conjugates of  $\kappa ={\kappa }^{(1)}$,  then

$$\text{N}(\kappa )=\underset{\text{integer}}{\underbrace{{\kappa }^{(1)}}}\underset{\text{integer}}{\underbrace{{\kappa }^{(2)}\mathrm{\cdots }{\kappa }^{(n)}}}=\kappa {\kappa }^{\prime }\in \mathbb{Z}.$$

Hence, $\kappa$ divides $\text{N}(\kappa )$ in the ring of integers of $K$, and (2) implies

$$\omega \equiv \varrho (mod\kappa ).$$

Since any number $r_i$ has $|\text{N}(\kappa )|$ different possible values $0,\mathrm{\hspace{0.17em}1},\mathrm{\dots },|\text{N}(\kappa )|-1$, there exist ${|\text{N}(\kappa )|}^n$ different ordered tuplets  $(r_1,r_2,\mathrm{\dots },r_n)$.  Therefore there exist at most${|\text{N}(\kappa )|}^n$ different residues and residue classes in the ring.