| Version current |
Version 11 |
| \begin{thmplain} |
\begin{thmplain} |
| \, Let $K$ be an algebraic number field and $\alpha$ an element of $K$.\, The norm $\mbox{N}(\alpha)$ and the trace $\mbox{S}(\alpha)$ of $\alpha$ in the field extension $K/\mathbb{Q}$ both are rational numbers and especially rational integers in the case $\alpha$ is an algebraic integer.\, If $\beta$ is another element of $K$, then |
\, Let $K$ be an algebraic number field and $\alpha$ an element of $K$.\, The norm $\mbox{N}(\alpha)$ and the trace $\mbox{S}(\alpha)$ of $\alpha$ in the field extension $K/\mathbb{Q}$ both are rational numbers and especially rational integers in the case $\alpha$ is an algebraic integer.\, If $\beta$ is another element of $K$, then |
| \begin{align} |
\begin{align} |
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\mbox{N}(\alpha\beta) \;=\; \mbox{N}(\alpha)\mbox{N}(\beta), \quad
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\mbox{N}(\alpha\beta) = \mbox{N}(\alpha)\mbox{N}(\beta), \quad
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\mbox{S}(\alpha\!+\!\beta) \;=\; \mbox{S}(\alpha)\!+\!\mbox{S}(\beta),
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\mbox{S}(\alpha+\beta) = \mbox{S}(\alpha)+\mbox{S}(\beta),
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| \end{align} |
\end{align} |
| i.e. the norm is multiplicative and the trace additive.\, If\, |
i.e. the norm is multiplicative and the trace additive.\, If\, |
| $[K\!:\!\mathbb{Q}] = n$\, and\, $a\in\mathbb{Q}$, then |
$[K\!:\!\mathbb{Q}] = n$\, and\, $a\in\mathbb{Q}$, then |
| $$\mbox{N}(a) = a^n, \quad \mbox{S}(a) = na.$$ |
$$\mbox{N}(a) = a^n, \quad \mbox{S}(a) = na.$$ |
| \end{thmplain} |
\end{thmplain} |
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| \textbf{Remarks} |
\textbf{Remarks} |
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| 1.\, The notions norm and trace were originally introduced in German \PMlinkescapetext{language} as ``die Norm'' and ``die Spur''.\, Therefore in German and many other literature the symbol of trace is S, Sp or sp.\, Nowadays the symbols T and Tr are common. |
1.\, The notions norm and trace were originally introduced in German \PMlinkescapetext{language} as ``die Norm'' and ``die Spur''.\, Therefore in German and many other literature the symbol of trace is S, Sp or sp.\, Nowadays the symbols T and Tr are common. |
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| 2.\, The norm and trace of an algebraic number $\alpha$ in the field extension\, $\mathbb{Q}(\alpha)/\mathbb{Q}$,\, i.e. the product and sum of all algebraic conjugates of $\alpha$, are called the {\em absolute norm} and the {\em absolute trace} of $\alpha$.\, Formulae like (1) concerning the absolute norms and traces are not sensible.\, |
2.\, The norm and trace of an algebraic number $\alpha$ in the field extension\, $\mathbb{Q}(\alpha)/\mathbb{Q}$,\, i.e. the product and sum of all algebraic conjugates of $\alpha$, are called the {\em absolute norm} and the {\em absolute trace} of $\alpha$.\, Formulae like (1) concerning the absolute norms and traces are not sensible.\, |
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| \begin{thmplain} |
\begin{thmplain} |
| \, An algebraic integer $\varepsilon$ is a unit if and only if |
\, An algebraic integer $\varepsilon$ is a unit if and only if |
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$$\mbox{N}(\varepsilon) \;=\; \pm 1,$$
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$$\mbox{N}(\varepsilon) = \pm 1,$$
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| i.e. iff the absolute norm of $\varepsilon$ is a rational unit.\, Thus \PMlinkescapetext{the constant term} in the minimal polynomial of an algebraic unit is always \,$\pm 1$. |
i.e. iff the absolute norm of $\varepsilon$ is a rational unit.\, Thus \PMlinkescapetext{the constant term} in the minimal polynomial of an algebraic unit is always \,$\pm 1$. |
| \end{thmplain} |
\end{thmplain} |
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\textbf{Example.}\, The minimal polynomial of the number $2\!+\!\sqrt{3}$, which is the fundamental unit of the quadratic field $\mathbb{Q}(\sqrt{3})$, is \,$x^2\!-\!4x\!+\!1$.
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\textbf{Example.}\, The minimal polynomial of the number $2+\sqrt{3}$, which is the fundamental unit of the quadratic field $\mathbb{Q}(\sqrt{3})$, is \,$x^2-4x+1$.
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