Theorem 1   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

N   NN   S   SS

(1)

i.e. the norm is multiplicative and the trace additive. If $[K\!:\!\mathbb{Q}] = n$ and $a\in\mathbb{Q}$ , then $$\mbox{N}(a) = a^n, \quad \mbox{S}(a) = na.$$

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.

Example. The minimal polynomial of the number $2\!+\!\sqrt{3}$ , which is the fundamental unit of the quadratic field $\mathbb{Q}(\sqrt{3})$ , is $x^2\!-\!4x\!+\!1$ .