# Sets closed under complex exponentiation

1. Jan 30, 2012

### alexfloo

The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and rational exponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, $(-1)^i=e^{-\pi}$, with is not rational, and in fact it is even transcendental.

Is there any algebraic theory that studies number fields that are closed under complex exponentiation, or otherwise the conditions under which an exponent of rational/algebraic numbers is itself algebraic?

2. Jan 30, 2012

### micromass

Here is a possible start for your investigations: http://en.wikipedia.org/wiki/Closed-form_expression

It appears that the Liouville numbers are the smallest algebraically closed field that is closed under both exponentiation and logarithm. (If you wiki "Liouville numbers" then you end up with a different concept however).