degree of algebraic number

Theorem.  The degree ( of any algebraic numberMathworldPlanetmath α in the number fieldMathworldPlanetmath (ϑ) divides the degree of ϑ.  The zeroes of the characteristic polynomialMathworldPlanetmathPlanetmath g(x) of α consist of the algebraic conjugates of α, each of which having equal multiplicityMathworldPlanetmath as zero of g(x).

Proof.  Let the minimal polynomialPlanetmathPlanetmath of  α  be


and all zeroes of this be  α1=α,α2,,αk.  Denote the canonical polynomial of α with respect to the primitive elementMathworldPlanetmathPlanetmath ( ϑ by r(x); then

a(r(ϑ))=a(α)= 0.

If  a(r(x)):=φ(x),  then the equation

φ(x)= 0

has rational coefficients and is satisfied by ϑ.  Since the minimal polynomial f(x) of ϑ is irreduciblePlanetmathPlanetmath (, it must divide φ(x) and all algebraic conjugates  ϑ1=ϑ,ϑ2,,ϑn of ϑ  make φ(x) zero.  Hence we have

a(α(i))=a(r(ϑi))= 0fori= 1, 2,,n

where the numbers α(i) are the (ϑ)-conjugatesPlanetmathPlanetmathPlanetmath ( of α.  Thus these (ϑ)-conjugates are roots of the irreducible equation  a(x)=0, whence a(x) must divide the characteristic polynomial g(x).  Let the power ( [a(x)]m exactly divide g(x), when


Antithesis:  deg(b(x)) 1    and   b(β)= 0.
This implies that  g(β)=0,  i.e. β is one of the numbers α(i).  Therefore, β were a zero of a(x) and thus a(x)b(x), which is impossible.  Consequently,the antithesis is wrong, i.e. b(x) is a constant, which must be 1 because g(x) and a(x) are monic polynomials.  So,  g(x)=[a(x)]m.  Since


it follows that


Hence  km=n  and k divides n, as asserted.  Moreover, each αj is a zero of order m of g(x), i.e. appears among the roots α(1),α(2),,α(n) of the equation  g(x)=0m times.

Title degree of algebraic numberPlanetmathPlanetmath
Canonical name DegreeOfAlgebraicNumber
Date of creation 2013-03-22 19:08:51
Last modified on 2013-03-22 19:08:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Classification msc 11C08
Classification msc 12F05
Classification msc 12E05
Related topic KConjugates
Related topic CharacteristicPolynomialOfAlgebraicNumber