Closure of Integers under multiplication and rational exponentiation

Question

What is the closure of the Integers under a finite number of multiplications and rational exponentiations?

For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this closure.

Are there any complex numbers provably not in this closure? Can you describe the entire set with a simple, closed-form solution?

Answer 1

Suppose we are given real number $r$ and let $0\lt \epsilon \lt 1$ be given. Assume that there is a function $z=\log_y x$ for real numbers $x,y$ which gives the exact real number $z$ such that $y^z=x$. Let $k=\log_2 r$. Choose $p\over q$ as a rational approximation of $k$ such that $\epsilon\lt \dfrac{2^{p\over q}}r\lt\frac 1\epsilon$.

By this method, it is clear that any real number can be approximated as nearly as desired. By similar mechanisms, we can approximate any complex number as nearly as we wish by multiplying by a root of unity.

Answer 2

Given your $1^{1/3}$ example, I think that the set you describe is the set of all complex numbers of the form $\omega a^{1/m}$, where $\omega$ is a root of unit and $a$ and $m$ are positive integers.

This can be proved by induction on the complexity of a number, which is the smallest number of elementary steps (multiplication and rational exponentiaton) required to produce the number.