# 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.

• $\mathrm{N}(\mathfrak{ab})=\mathrm{N}(\mathfrak{a})\!\cdot\!\mathrm{N}(% \mathfrak{b})$

• $\mathrm{N}(\mathfrak{a})=1\quad\Leftrightarrow\quad\mathfrak{a}=(1)$

• $\mathrm{N}((\alpha))=|\mathrm{N}(\alpha)|$

• $\mathrm{N}(\mathfrak{a})\in\mathfrak{a}$

• 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