| Version 7 |
Version 6 |
| 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 {\em congruent modulo the ideal} $\mathfrak{m}$ in the case that\, $\alpha\!-\!\beta \in \mathfrak{m}$.\, This is denoted by |
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 {\em congruent modulo the ideal} $\mathfrak{m}$ in the case that\, $\alpha\!-\!\beta \in \mathfrak{m}$.\, This is denoted by |
| $$\alpha \equiv \beta \pmod{\mathfrak{m}}.$$ |
$$\alpha \equiv \beta \pmod{\mathfrak{m}}.$$ |
| This congruence relation \PMlinkescapetext{divides} the ring of integers of $K$ into equivalence classes, which are called the {\em residue classes modulo the ideal} $\mathfrak{m}$. |
This congruence relation \PMlinkescapetext{divides} the ring of integers of $K$ into equivalence classes, which are called the {\em residue classes modulo the ideal} $\mathfrak{m}$. |
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| \textbf{Definition.}\, Let $K$ be an algebraic number field and\, $\mathfrak{a}$\, a non-zero ideal in $K$.\, The {\em absolute norm of ideal} $\mathfrak{a}$ means the number of all residue classes modulo $\mathfrak{a}$. |
\textbf{Definition.}\, Let $K$ be an algebraic number field and\, $\mathfrak{a}$\, a non-zero ideal in $K$.\, The {\em absolute norm of ideal} $\mathfrak{a}$ means the number of all residue classes modulo $\mathfrak{a}$. |
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| \textbf{Remark.}\, The norm of any ideal $\mathfrak{a}$ of $K$ is finite --- it has the expression |
\textbf{Remark.}\, The norm of any ideal $\mathfrak{a}$ of $K$ is finite --- it has the expression |
| $$\mbox{N}(\mathfrak{a}) = \sqrt{\frac{\Delta(\mathfrak{a})}{d}}$$ |
$$\mbox{N}(\mathfrak{a}) = \sqrt{\frac{\Delta(\mathfrak{a})}{d}}$$ |
| where $\Delta(\mathfrak{a})$ is the discriminant of the ideal and $d$ the fundamental number of the field. |
where $\Delta(\mathfrak{a})$ is the discriminant of the ideal and $d$ the fundamental number of the field. |
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| \PMlinkescapetext{\textbf{Some properties}} |
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| \begin{itemize} |
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| \item $\mbox{N}(\mathfrak{ab}) |
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| = \mbox{N}(\mathfrak{a})\!\cdot\!\mbox{N}(\mathfrak{b})$ |
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| \item $\mbox{N}(\mathfrak{a}) = 1 \quad\Leftrightarrow\quad \mathfrak{a} = (1)$ |
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| \item $\mbox{N}((\alpha)) = |\mbox{N}(\alpha)|$ |
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| \item $\mbox{N}(\mathfrak{a}) \in \mathfrak{a}$ |
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| \item If $\mbox{N}(\mathfrak{p})$ is a rational prime, then $\mathfrak{p}$ is a prime ideal. |
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| \end{itemize} |
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