Question 64
Let $n \ge 1$ be a positive integer and for all $t \in \mathbb{R}$ let
$$
P(t) = 1 + t + t^2 + \dots + t^{2n}
$$
If $x \in \mathbb{R}$, $P(x) \in \mathbb{Q}$ and $P(x^2) \in \mathbb{Q}$, then show that $x \in \mathbb{Q}$.
$$
P(t) = 1 + t + t^2 + \dots + t^{2n}
$$
If $x \in \mathbb{R}$, $P(x) \in \mathbb{Q}$ and $P(x^2) \in \mathbb{Q}$, then show that $x \in \mathbb{Q}$.