norm and trace of algebraic number

Theorem 1.

Let K be an algebraic number fieldMathworldPlanetmath and α an element of K.  The norm N(α) and the trace S(α) of α in the field extension K/ both are rational numbers and especially rational integers in the case α is an algebraic integerMathworldPlanetmath.  If β is another element of K, then

N(αβ)=N(α)N(β),S(α+β)=S(α)+S(β), (1)

i.e. the norm is multiplicative and the trace additivePlanetmathPlanetmath.  If  [K:]=n  and  a, then



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 numberMathworldPlanetmath α in the field extension  (α)/,  i.e. the productPlanetmathPlanetmathPlanetmath and sum of all algebraic conjugates of α, are called the absolute norm and the absolute trace of α.  Formulae like (1) concerning the absolute norms and traces are not sensible.

Theorem 2.

An algebraic integer ε is a unit if and only if


i.e. iff the absolute norm of ε is a rational unit.  Thus in the minimal polynomial of an algebraic unit is always  ±1.

Example.  The minimal polynomial of the number 2+3, which is the fundamental unitMathworldPlanetmath of the quadratic field (3), is  x2-4x+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