PlanetMath: ideal norm

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ideal norm

(Definition)

Let $\alpha$ and $\beta$ be algebraic integers in an algebraic number field and $\mathfrak{m}$ a non-zero ideal in the ring of integers of . 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

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

This congruence relation divides the ring of integers of

into equivalence classes, which are called the residue classes modulo the ideal $\mathfrak{m}$ .

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

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

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

where $\Delta(\mathfrak{a})$ is the discriminant of the ideal and

the fundamental number of the field.

Some properties

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

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"ideal norm" is owned by pahio. [ full author list (2) ]

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Other names:

norm of an ideal, norm of ideal

Also defines:

congruent modulo the ideal, residue classes modulo the ideal, absolute norm of ideal

Keywords:

residue class, congruence

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Cross-references: prime ideal, rational prime, field, fundamental number, discriminant of the ideal, expression, finite, residue classes, number, equivalence classes, congruence relation, ring of integers, ideal, algebraic number field, algebraic integers
There are 3 references to this entry.

This is version 11 of ideal norm, born on 2006-03-04, modified 2008-02-21.
Object id is 7672, canonical name is IdealNorm.
Accessed 4116 times total.

Classification:

AMS MSC:

11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)

Pending Errata and Addenda

None.

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