| Version 3 |
Version 2 |
| \begin{thmplain} |
\begin{thmplain} |
| \, Let $\alpha$ be an algebraic number. \,The norm $\mbox{N}(\alpha)$, i.e. the product of all algebraic conjugates of $\alpha$, and the trace $\mbox{S}(\alpha)$, i.e. the sum of the algebraic conjugates of $\alpha$, both are rational numbers, and especially rational integers in the case $\alpha$ is an algebraic integer. |
\, Let $\alpha$ be an algebraic number. \,The norm $\mbox{N}(\alpha)$, i.e. the product of all algebraic conjugates of $\alpha$, and the trace $\mbox{S}(\alpha)$, i.e. the sum of the algebraic conjugates of $\alpha$, both are rational numbers, and especially rational integers in the case $\alpha$ is an algebraic integer. |
| If $\beta$ is another algebraic number, then |
If $\beta$ is another algebraic number, then |
| $$\mbox{N}(\alpha\beta) = \mbox{N}(\alpha)\mbox{N}(\beta), \quad |
$$\mbox{N}(\alpha\beta) = \mbox{N}(\alpha)\mbox{N}(\beta), \quad |
| \mbox{S}(\alpha+\beta) = \mbox{S}(\alpha)+\mbox{S}(\beta),$$ |
\mbox{S}(\alpha+\beta) = \mbox{S}(\alpha)+\mbox{S}(\beta),$$ |
| i.e. the norm is multiplicative and the trace additive. |
i.e. the norm is multiplicative and the trace additive. |
| \end{thmplain} |
\end{thmplain} |
|
|
| \begin{thmplain} |
\begin{thmplain} |
|
\, An algebraic integer $\varepsilon$ is a unit if anf only if its norm is\,
|
\, An algebraic integer $\varepsilon$ is a unit if anf only if its norm is
|
|
$\pm 1$. \,Thus the constant term in the minimal polynomial of an algebraic unit is always \,$\pm 1$.
|
$\pm 1$.$\pm 1$.
|
| \end{thmplain} |
\end{thmplain} |
