Let $F$ be an ordered field and let $R$ denote its real closure.

It is well-known that $F$ is cofinal in $R$, but not necessarily dense. For example, consider $F=\mathbb{R}(\omega)$ with the order determined by $r<\omega$ for all $r\in \mathbb{R}$, and consider the open interval $(\sqrt{\omega}-1,\sqrt{\omega}+1)$ in $R$, which does not meet $F$.

Question 1.Is always the case that radicals of elements of $F$ in $R$ are dense in $R$? Equivalently, is it possible to find for all $0<\alpha<\beta$ in $R$ an element $\gamma\in F$ and some $n\in \mathbb{N}$ such that $\alpha^n<\gamma<\beta^n$?

It is not difficult to see that a positive answer would imply a positive answer to the next question, which is really what I am after.

Question 2.Let $a_0,a_1,\dots,a_t\in R$ be distinct elements. Is there always a polynomial $p\in F[x]$ satisfying $p(a_0)<0$ while $p(a_i)>0$ for all $i\in\{1,\dots,t\}$?

The most interesting case is when $a_0,\dots,a_t$ constitute the list of roots of an irreducible polynomial in $F[x]$.