norm and trace of algebraic number

Theorem 1.

Let $K$ be an algebraic number field and $\alpha$ an element of $K$.  The norm $\text{N}(\alpha )$ and the trace $\text{S}(\alpha )$ 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

$\text{N}(\alpha \beta )=\text{N}(\alpha )\text{N}(\beta ),\text{S}(\alpha +\beta )=\text{S}(\alpha )+\text{S}(\beta ),$(1)

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

$$\text{N}(a)=a^n,\text{S}(a)=na.$$

Remarks

1.  The notions norm and trace were originally introduced in German 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 $\epsilon$ is a unit if and only if

$$\text{N}(\epsilon )=\pm 1,$$

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

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

Title	norm and trace of algebraic number
Canonical name	NormAndTraceOfAlgebraicNumber
Date of creation	2013-03-22 15:19:08
Last modified on	2013-03-22 15:19:08
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11R04
Related topic	TheoryOfAlgebraicNumbers
Related topic	AlgebraicNumberTheory
Related topic	IdealNorm
Related topic	UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding
Related topic	IndependenceOfCharacteristicPolynomialOnPrimitiveElement
Defines	absolute norm
Defines	absolute trace