transcendental root theorem

Suppose a constant x is transcendental over some field F. Then xn is also transcendental over F for any n1.


Let F¯ denote an algebraic closureMathworldPlanetmath of F. Assume for the sake of contradictionMathworldPlanetmathPlanetmath that xnF¯. Then since algebraic numbersMathworldPlanetmath are closed under multiplicationPlanetmathPlanetmath (and thus exponentiation by positive integers), we have (xn)n=xF¯, so that x is algebraic over F, creating a contradiction. ∎

Title transcendental root theorem
Canonical name TranscendentalRootTheorem
Date of creation 2013-03-22 14:04:23
Last modified on 2013-03-22 14:04:23
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type TheoremMathworldPlanetmath
Classification msc 11R04