August 23rd, 2017, 06:29 AM  #1 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1  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. 
August 23rd, 2017, 07:54 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 112 Thanks: 35 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
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.

August 23rd, 2017, 08:11 AM  #3 
Member Joined: May 2017 From: France Posts: 57 Thanks: 1 
I know a solution, but I like proposed mathematics enigma.


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