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P(x), P(x^2) are rational then x is rational, where P(t)=1+t +t^2 +...++ t^{2n}

Source: IMAC Arhimede 2014 p4

May 6, 2019

polynomialalgebraSum of powersSumrational

Problem Statement

Let

n

be a natural number and let

P (t) = 1 + t + t^2 + ... + t^{2n}

. If

x \in R

such that

P (x)

and

P (x^2)

are rational numbers, prove that

x

is rational number.