independence of characteristic polynomial on primitive element

The simple field extension (ϑ)/ where ϑ is an algebraic numberMathworldPlanetmath of degree ( n may be determined also by using another primitive elementMathworldPlanetmath η.  Then we have


whence, by the entry degree of algebraic numberPlanetmathPlanetmath, the degree of η divides the degree of ϑ.  But also


whence the degree of ϑ divides the degree of η.  Therefore any possible primitive element of the field extension has the same degree n.  This number is the degree of the number fieldMathworldPlanetmath (, i.e. the degree of the field extension, as comes clear from the entry canonical form of element of number field.

Although the characteristic polynomialMathworldPlanetmathPlanetmath


of an element α of the algebraic number field (ϑ) is based on the primitive element ϑ, the equation

g(x)=(x-α1)m(x-α2)m(x-αk)m (1)

in the entry of algebraic number shows that the polynomialPlanetmathPlanetmath is fully determined by the algebraic conjugates of α itself and the number m which equals the degree n divided by the degree k of α.

The above stated makes meaningful to define the norm and the trace functions in an algebraic number field as follows.

Definition.  If α is an element of the number field (ϑ), then the norm N(α) and the trace S(α) of α are the product and the sum, respectively, of allϑ)-conjugatesPlanetmathPlanetmath α(i) of α.

Since the coefficients of the characteristic equation of α are rational, one has


In fact, one can infer from (1) that

N(α)=akm,S(α)=-ma1, (2)

where xk+a1xk-1++ak is the minimal polynomialPlanetmathPlanetmath of α.

Title independence of characteristic polynomial on primitive element
Canonical name IndependenceOfCharacteristicPolynomialOnPrimitiveElement
Date of creation 2014-02-04 8:07:18
Last modified on 2014-02-04 8:07:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Topic
Classification msc 11R04
Classification msc 12F05
Classification msc 11C08
Classification msc 12E05
Synonym norm and trace functions in number field
Related topic Norm
Related topic NormAndTraceOfAlgebraicNumber
Related topic PropertiesOfMathbbQvarthetaConjugates
Related topic DiscriminantInAlgebraicNumberField
Defines norm in number field
Defines trace in number field
Defines norm
Defines trace