nAalniaOMliO wrote: Does there exist a polynomial $p(x)$ with integer coefficients for which $$p(\sqrt{2})=\sqrt{2}$$$$p(2\sqrt{2})=2\sqrt{2}+2$$ Writing $P(x)=Q(x^2)+xR(x^2)$, we get $Q(2)=0$ and $Q(8)=2$, which is impossible since then it would imply $(8-2)|(2-0)$, which is wrong.