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August 23rd, 2017, 06:29 AM   #1
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Irrationnal field

Hi,

$\displaystyle P\in \mathbb C [x] \text{ with }P(\mathbb Q(\sqrt 2)) \subset \mathbb Q(\sqrt 2).
\\\text{Is it true that : }P \in \mathbb Q(\sqrt 2)[x] ?$

Cordially.
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August 23rd, 2017, 07:54 AM   #2
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It's true and can be proven quite easily by induction on the degree of $\displaystyle P$. I'd suggest thinking about it a bit further, but if you're still stuck after that, I'll be happy to give hints.
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August 23rd, 2017, 08:11 AM   #3
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I know a solution, but I like proposed mathematics enigma.
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