Is there a polynomial whose Coefficients in $\mathbb Q$ such that none of roots of this polynomial be in the root closure of $\mathbb Q$?

Question

For example, we know that $x^5-x+1$ is a polynomial of degree five over $\mathbb Q$ that is not solvable by radicals. Which means that one of the roots of this polynomial is not in root closure of $\mathbb Q$. Is there a polynomial whose coefficients in $\mathbb Q$ such that none of the roots of this polynomial be in the root closure of $\mathbb Q$? In the other hand, can we find a polynomial over $\mathbb Q$ such that none of the roots of this polynomial Obtained by the operations multiplying, suming, Subtracting, Divising and rooting from elements of $\mathbb Q$