Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$ ?
I have tried the thought that to lower its degree as below.
Let $f(x)=\alpha_n x^n+...+\alpha_1x+\alpha_0$. Since $0\in F$, $f(0)=\alpha_0 \in F$.
$(f(x)-\alpha_0)/x\in F$ when $x\neq 0$. Hence, $\alpha_n x^{n-1}+...+\alpha_1\in F$ when $x\neq 0$.
I want to say that $\alpha_1\in F$ in a similar way. However, the restriction that $x\neq 0$ does not allow me to claim it.
I believe that this proposition is correct but I cannot prove it. I need some help or a simple hint to get myself out. Thanks.
Edit. $\alpha_n=1$ in $f(x)$