I need to prove or disprove the following claim. Let $ x \notin \Bbb Q $ such that $ x^3 \in \Bbb Q $. Then $x^2+x+1 \notin \Bbb Q$. I tried to find a lot of counter examples in order to disprove it, yet couldn't find anything. I was also unable to prove it. Any assistance will be welcomed. Thank you very much.
2026-04-08 18:06:26.1775671586
Prove or disprove a claim regarding rational numbers
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Suppose $x^2+x+1 \in \Bbb Q$. Because $x^3 \in \Bbb Q$, so is $x^3-1$.
We know $x^3-1 = (x-1)(x^2+x+1) $ (check it!), so $\frac{x^3-1}{x^2+x+1} = x-1 \in \Bbb Q$. Therefore $x \in \Bbb Q$. Which is a contradiction.
Note that $x^2+x+1=0$ has no real solution.