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单词 TranscendentalRootTheorem
释义

transcendental root theorem


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

Proof.

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.∎

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