characteristic polynomial of algebraic number

Let ϑ be an algebraic numberMathworldPlanetmath of degree n, f(x) its minimal polynomialPlanetmathPlanetmath and


its algebraic conjugates.

Let α be an element of the number fieldMathworldPlanetmath (ϑ) and


the canonical polynomial of α with respect to ϑ.  We consider the numbers

r(ϑ1)=α:=α(1),r(ϑ2):=α(2),,r(ϑn):=α(n) (1)

and form the equation

g(x):=i=1n[x-r(ϑi)]=(x-α(1))(x-α(2))(x-α(n))=xn+g1xn-1++gn= 0,

the roots of which are the numbers (1) and only these.  The coefficientsMathworldPlanetmath gi of the polynomialMathworldPlanetmathPlanetmath g(x) are symmetric polynomialsMathworldPlanetmath in the numbers ϑ1,ϑ2,,ϑn and also symmetric polynomials in the numbers α(i).  The fundamental theorem of symmetric polynomials implies now that the symmetric polynomials gi in the roots ϑi of the equation  f(x)=0  belong to the ring determined by the coefficients of the equation and of the canonical polynomial r(x); thus the numbers gi are rational (whence the degree of α is at most equal to n).

It is not hard to show (see the entry degree of algebraic numberPlanetmathPlanetmath) of that the degree k of α divides n and that the numbers (1) consist of α and its algebraic conjugates α2,,αk, each of which appears in (1) exactly  nk=m  times.  In fact,  g(x)=[a(x)]m  where a(x) is the minimal polynomial of α (consequently, the coefficients gi are integers if α is an algebraic integerMathworldPlanetmath).

The polynomial g(x) is the characteristic polynomialMathworldPlanetmathPlanetmath (in German Hauptpolynom) of the element α of the algebraic number field (ϑ) and the equation  g(x)=0 the characteristic equation (Hauptgleichung) of α.  See the independence of characteristic polynomial on primitive elementMathworldPlanetmathPlanetmath.

So, the roots of the characteristic equation of α are α(1),α(2),,α(n).  They are called the (ϑ)-conjugatesPlanetmathPlanetmath of α; they all are algebraic conjugates of α.

Title characteristic polynomial of algebraic number
Canonical name CharacteristicPolynomialOfAlgebraicNumber
Date of creation 2013-03-22 19:08:41
Last modified on 2013-03-22 19:08:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 15A18
Classification msc 12F05
Classification msc 11R04
Related topic RationalIntegersInIdeals
Related topic DegreeOfAlgebraicNumber
Defines characteristic polynomial
Defines characteristic equation
Defines (ϑ)-conjugates
Defines K-conjugates