Is there a name for a radical field extension of the rationals that contains all radical/solvable elements?

Question

I am wondering if there exists a name for the field $F$ such that $\mathbb{Q}\subset F\subset \mathbb{A}$, and $F$ contains all the radical elements such as $\sqrt[7]{2}, \sqrt[3]{3-\sqrt[4]{7}}, \sqrt[3]{1+\sqrt{4-\sqrt[5]{2}}}$, but not unsolvable elements such as the roots of $x^5+x+1$ ?