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Let $p_1,\ldots,p_n$ be a sequence of distinct primes $>0$. How to prove that $\sqrt{p_n}\not\in \mathbb{Q}(\sqrt{p_1}, \ldots, \sqrt{p_{n-1}})$. I tried to prove it be the principle of the mathematical induction but i could not. Also, i could not complete a proof by contradiction. I will be grateful if someone helps me solving that.

Thanks in advance.

Hussein Eid
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  • Another previous Question can be used to arrive at the conclusion called for here. I'm a bit concerned that this implication is worthy of some exposition. But I'll leave it to the OP to study these previous posts (and their Answers) and speak up if such exposition is desired. – hardmath Dec 28 '19 at 05:00

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