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'norm and trace of algebraic number'
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| Title of object: |
norm and trace of algebraic number |
| Canonical Name: |
NormAndTraceOfAlgebraicNumber |
| Type: |
Theorem |
| Created on: |
2005-05-29 17:19:24 |
| Modified on: |
2005-05-29 17:43:18 |
| Classification: |
msc:11R04 |
Preamble:
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\newtheorem{thmplain}{Theorem} |
Content:
\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.
If $\beta$ is another algebraic number, then
$$\mbox{N}(\alpha\beta) = \mbox{N}(\alpha)\mbox{N}(\beta), \quad
\mbox{S}(\alpha+\beta) = \mbox{S}(\alpha)+\mbox{S}(\beta),$$
i.e. the norm is multiplicative and the trace additive.
\end{thmplain}
\begin{thmplain}
\, 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 \,$\pm 1$.
\end{thmplain} |
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