PlanetMath: norm and trace of algebraic number

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norm and trace of algebraic number

(Theorem)

Theorem 1 Let

be an algebraic number field and $\alpha$ an element of

. The norm N $(\alpha)$ and the trace 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

, then

N $\displaystyle (\alpha\beta) =$ N $\displaystyle (\alpha)$ N $\displaystyle (\beta),$ S $\displaystyle (\alpha+\beta) =$ S $\displaystyle (\alpha)+$ S $\displaystyle (\beta),$

(1)

i.e. the norm is multiplicative and the trace additive. If $[K\!:\!\mathbb{Q}] = n$ and $a\in\mathbb{Q}$ , then

N $\displaystyle (a) = a^n,$ S $\displaystyle (a) = na.$

Remarks

1. The notions norm and trace were originally introduced in German 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.

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 absolute norm and the absolute trace of $\alpha$ . Formulae like (1) concerning the absolute norms and traces are not sensible.

Theorem 2 An algebraic integer $\varepsilon$ is a unit if and only if

N $\displaystyle (\varepsilon) = \pm 1,$

i.e. iff the absolute norm of $\varepsilon$ is a rational unit. Thus the constant term in the minimal polynomial of an algebraic unit is always $\pm 1$ .