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, 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 3849 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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