proof of transcendental root theorem

Proposition 1.

Let FK be a field extension with K an algebraically closed field. Let xK be transcendental over F. Then for any natural numberMathworldPlanetmath n1, the element x1/nK is also transcendental over F.


Suppose x is transcendental over a field F, and assume for a contradictionMathworldPlanetmathPlanetmath that x1/n is algebraic over F. Thus, there is a polynomialMathworldPlanetmathPlanetmathPlanetmath P(y)F[y] such that P(x1/n)=0 (note that the polynomial yn-x is not a polynomial with coefficients in F, so P(y) might be more involved). Then the field H=F(x1/n)K is a finite algebraic extensionMathworldPlanetmath of F, and every element of H is algebraic over K. However xH, so x is algebraic over F which is a contradiction. ∎

Title proof of transcendental root theorem
Canonical name ProofOfTranscendentalRootTheorem
Date of creation 2013-03-22 14:11:41
Last modified on 2013-03-22 14:11:41
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Proof
Classification msc 11R04
Related topic AlgebraicElement
Related topic AlgebraicClosure
Related topic Algebraic
Related topic AlgebraicExtension
Related topic AFiniteExtensionOfFieldsIsAnAlgebraicExtension