# congruence in algebraic number field

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

 $\displaystyle\alpha\equiv\beta\pmod{\kappa}$ (1)

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

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

 $\displaystyle\omega=\mbox{N}(\kappa)\pi\!+\!\varrho,$ (2)

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

 $\mbox{N}(\kappa)=\underbrace{\kappa^{(1)}}_{\mbox{integer}}\underbrace{\kappa^% {(2)}\cdots\kappa^{(n)}}_{\mbox{integer}}=\kappa\kappa^{\prime}\in\mathbb{Z}.$

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

 $\omega\equiv\varrho\pmod{\kappa}.$

Since any number $r_{i}$ has $|\mbox{N}(\kappa)|$ different possible values $0,\,1,\,\ldots,\,|\mbox{N}(\kappa)|\!-\!1$, there exist $|\mbox{N}(\kappa)|^{n}$ different ordered tuplets$(r_{1},\,r_{2},\,\ldots,\,r_{n})$.  Therefore there exist at most $|\mbox{N}(\kappa)|^{n}$ different residues and residue classes in the ring.

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