ideal norm

Let $\\alpha$ and $\\beta$ be algebraic integers in an algebraic number field $K$ and $\\mathfrak{m}$ a non-zero ideal in the ring of integers of $K$ . We say that $\\alpha$ and $\\beta$ are congruent modulo the ideal $\\mathfrak{m}$ in the case that $\\alpha\\!-\\!\\beta\\in\\mathfrak{m}$ . This is denoted by

\\alpha\\equiv\\beta\\pmod{\\mathfrak{m}}.

This congruence relation the ring of integers of $K$ into equivalence classes, which are called the residue classes modulo the ideal $\\mathfrak{m}$ .

Definition. Let $K$ be an algebraic number field and $\\mathfrak{a}$ a non-zero ideal in $K$ . The absolute norm of ideal $\\mathfrak{a}$ means the number of all residue classes modulo $\\mathfrak{a}$ .

Remark. The of any ideal $\\mathfrak{a}$ of $K$ is finite — it has the expression

\\mathrm{N}(\\mathfrak{a})=\\sqrt{\\frac{\\Delta(\\mathfrak{a})}{d}}

where $\\Delta(\\mathfrak{a})$ is the discriminant of the ideal and $d$ the fundamental number of the field.

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$\\mathrm{N}(\\mathfrak{ab})=\\mathrm{N}(\\mathfrak{a})\\!\\cdot\\!\\mathrm{N}(%\\mathfrak{b})$
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$\\mathrm{N}(\\mathfrak{a})=1\\quad\\Leftrightarrow\\quad\\mathfrak{a}=(1)$
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$\\mathrm{N}((\\alpha))=|\\mathrm{N}(\\alpha)|$
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$\\mathrm{N}(\\mathfrak{a})\\in\\mathfrak{a}$
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If $\\mathrm{N}(\\mathfrak{p})$ is a rational prime, then $\\mathfrak{p}$ 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