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 divides 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 norm of any ideal $\mathfrak{a}$ of $K$ is finite -- it has the expression $$\N(\mathfrak{a}) = \sqrt{\frac{\Delta(\mathfrak{a})}{d}}$$ where $\Delta(\mathfrak{a})$ is the discriminant of the ideal and $d$ the fundamental number of the field.

Some properties

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