ideal norm


Let Ξ± and Ξ² be algebraic integersMathworldPlanetmath in an algebraic number fieldMathworldPlanetmath K and π”ͺ a non-zero ideal in the ring of integers of K.  We say that Ξ± and Ξ² are congruent modulo the ideal π”ͺ in the case that  Ξ±-β∈π”ͺ.  This is denoted by

α≑β(modπ”ͺ).

This congruence relationPlanetmathPlanetmath the ring of integers of K into equivalence classesMathworldPlanetmathPlanetmath, which are called the residue classesMathworldPlanetmath modulo the ideal π”ͺ.

Definition.  Let K be an algebraic number field and  π”žβ€‰ a non-zero ideal in K.  The absolute norm of ideal π”ž means the number of all residue classes modulo π”ž.

Remark.  The of any ideal π”ž of K is finite β€” it has the expression

N⁒(π”ž)=Δ⁒(π”ž)d

where Δ⁒(π”ž) is the discriminantPlanetmathPlanetmathPlanetmath of the ideal and d the fundamental number of the field.

  • β€’

    N⁒(π”žβ’π”Ÿ)=N⁒(π”ž)β‹…N⁒(π”Ÿ)

  • β€’

    N⁒(π”ž)=1β€ƒβ‡”β€ƒπ”ž=(1)

  • β€’

    N⁒((α))=|N⁒(α)|

  • β€’

    N⁒(π”ž)βˆˆπ”ž

  • β€’

    If N⁒(𝔭) is a rational prime, then 𝔭 is a prime idealMathworldPlanetmathPlanetmathPlanetmath.

Title ideal norm
Canonical name IdealNorm
Date of creation 2013-03-22 15:43:23
Last modified on 2013-03-22 15:43:23
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Definition
Classification msc 11R04
Synonym norm of an ideal
Synonym norm of ideal
Related topic NormAndTraceOfAlgebraicNumber
Related topic CongruencesMathworldPlanetmath
Related topic MultiplicativeCongruence
Related topic BasisOfIdealInAlgebraicNumberField
Related topic IdealClassGroupIsFinite
Related topic RationalIntegersInIdeals
Defines congruence modulo an ideal
Defines congruent modulo the ideal
Defines residue classes modulo the ideal
Defines absolute norm of ideal