congruence in algebraic number field

Definition.  Let α, β and κ be integers ( of an algebraic number fieldMathworldPlanetmath K and  κ0.  One defines

αβ(modκ) (1)

if and only if  κα-β,  i.e. iff there is an integer λ of K with  α-β=λκ.

Theorem.  The congruenceMathworldPlanetmathPlanetmathPlanetmath” modulo κ defined above is an equivalence relationMathworldPlanetmath in the maximal orderPlanetmathPlanetmath of K.  There are only a finite amount of the equivalence classesMathworldPlanetmath, the residue classes modulo κ.

Proof.  For justifying the transitivity of “”, suppose (1) and  βγ(modκ); then there are the integers λ and μ of K such that  α-β=λκ,  β-γ=μκ.  Adding these equations we see that  α-γ=(λ+μ)κ  with the integer λ+μ of K.  Accordingly,  αγ(modκ).
Let ω be an arbitrary integer of K and  {ω1,ω2,,ωn}  a minimal basis of the field.  Then we can write


where the ai’s are rational integers.  For  i=1, 2,,n, the division algorithmPlanetmathPlanetmath determines the rational integers qi and ri with




So we have

ω=N(κ)π+ϱ, (2)

where π and ϱ are some integers of the field.  If  κ(1),κ(2),,κ(n)  are the algebraic conjugates of  κ=κ(1),  then


Hence, κ divides N(κ) in the ring of integers of K, and (2) implies


Since any number ri has |N(κ)| different possible values 0, 1,,|N(κ)|-1, there exist |N(κ)|n different ordered tuplets(r1,r2,,rn).  Therefore there exist at most |N(κ)|n different residues and residue classes in the ring.

Title congruence in algebraic number field
Canonical name CongruenceInAlgebraicNumberField
Date of creation 2013-03-22 18:17:11
Last modified on 2013-03-22 18:17:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 13B22
Synonym congruence in number field
Related topic CongruenceRelationOnAnAlgebraicSystem
Related topic ChineseRemainderTheoremInTermsOfDivisorTheory
Related topic Congruences
Defines residue class