I'm interested in the set $S$ of complex numbers that can be characterized as follows:
- $\mathbb{Z}$ is in $S$.
- If $a$ and $b$ are in $S$ then $a+b$, $a-b$ and $a*b$ are in $S$.
- If $a$ and $b$ are in $S$ and $b\neq{0}$ then $a/b$ is in $S$.
- If $a$ and $b$ are in $S$ and not both $0$ then $a^b$ is in $S$.
This set contains some numbers that are not algebraic, like $2^\sqrt{2}$ (by the Gelfond–Schneider theorem). It contains all algebraic numbers that are roots of solvable polynomials. Whether it contains the remaining algebraic numbers is not obvious to me.
My question is, has this set been studied much? Does it have a name, or belong to any particular field of math? I'm curious about whether, for example, denominator rationalization is always possible for these numbers, as it is with algebraic numbers, but I don't know where to start looking. I suppose there is also a question of whether it's even well-defined. I know exponentiation by complex numbers gets messy.