ideal norm
Let and be algebraic integers in an algebraic number field and a non-zero ideal in the ring of integers of .β We say that and are congruent modulo the ideal in the case thatβ .β This is denoted by
This congruence relation the ring of integers of into equivalence classes, which are called the residue classes modulo the ideal .
Definition.β Let be an algebraic number field andβ β a non-zero ideal in .β The absolute norm of ideal means the number of all residue classes modulo .
Remark.β The of any ideal of is finite β it has the expression
where is the discriminant of the ideal and the fundamental number of the field.
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β’
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β’
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β’
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β’
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If is a rational prime, then is a prime ideal.
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 | Congruences |
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 |