polynomial equation with algebraic coefficients

If α1,,αk are algebraic numbersMathworldPlanetmath [resp. algebraic integersMathworldPlanetmath] and


polynomialsMathworldPlanetmathPlanetmathPlanetmath in α1,,αk with rational [resp. integer] coefficients, then all complex rootsMathworldPlanetmath of the equation

xn+f1(α1,,αk)xn-1++fn(α1,,αk)= 0 (1)

are algebraic numbers [resp. algebraic integers].

Proof.  Let the minimal polynomial xm+a1xm-1++am of α1 over have the zeros (http://planetmath.org/ZeroOfAFunction)


and denote by  F(x;α1,α2,,αk)  the left hand side of the equation (1).  Consider the equation

G(x;α2,,αk):=i=1mF(x;α1(i),α2,,αk)= 0. (2)

Here, the coefficients of the polynomial G are polynomials in the numbers


with rational [resp. integer] coefficients.  Thus the coefficients of G are symmetric polynomialsMathworldPlanetmath gj in the numbers α1(i):


By the fundamental theorem of symmetric polynomials, the coefficients gj of G are polynomials in α2,,αk with rational [resp. integer] coefficients.  Consequently, G has the form


where the coefficients ai(α2,,αk) are polynoms in the numbers αj with rational [resp. integer] coefficients.  As one continues similarly, removing one by one also α2,,αk which go back to the rational [resp. integer] coefficients of the corresponding minimal polynomials, one shall finally arrive at an equation

xs+A1xs-1++As= 0, (3)

among the roots of which there are the roots of (1); the coefficients Aν do no more explicitely depend on the algebraic numbers α1,,αk but are rational numbers [resp. integers].
Accordingly, the roots of (1) are algebraic numbers [resp. algebraic integers], Q.E.D.

Title polynomial equation with algebraic coefficients
Canonical name PolynomialEquationWithAlgebraicCoefficients
Date of creation 2013-03-22 19:07:37
Last modified on 2013-03-22 19:07:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Synonym monic equation with algebraic coefficients