The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and(adsbygoogle = window.adsbygoogle || []).push({}); rationalexponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, [itex](-1)^i=e^{-\pi}[/itex], 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?

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# Sets closed under complex exponentiation

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