Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that
(1) $F(f)$ contains $f$ as a sub-field and
(2) $F(f)$ is closed under the operations of addition, subtraction, multiplication, division and extraction of $n$th roots (for any positive integer $n$)?
If $n$ is only allowed to have the value $2$, the resulting extension of $f$ is called the "Pythagorean closure" of $f$. So I am wondering if there is a name for $F(f)$ too, and if $F(f)$ plays a role in various geometrical constructions similar to the role played by Pythagorean closures in "straight edge and compass" constructions.
I believe I recall reading the name "solvable closure" for this construction, but Google doesn't seem to turn up anything. This would make sense, however, as $F(f)$ is the smallest extension of $f$ which is closed under the operation of taking solvable extensions. Alternatively, $\text{Gal}(F(f)/f)$ is the largest pro-solvable quotient of $\text{Gal}(\overline{f}/f)$.