Is it possible to compute complex powers in finite fields? Given a $\in \mathbb{F}\_p$ ($p$ prime), how can one compute $a^i$ per example?



Complex powers in finite fields
I certainly don't think that for arbitrary complex numbers, you'll be able to define a notion of exponentiation (take $\pi$ for example). However, I think if you restrict to algebraic numbers, you'll have more of a chance. Here's something I cooked up for (sometimes) handling an $n$th root of unity $\omega$:


I assume we would like exponentiation to $\omega$ to be a (group) automorphism of $\mathbb{F}\_q^\times\simeq\mathbb{Z}/(q-1)\mathbb{Z}$ - note that $\sigma(a)=a^\omega$ would have the property that $\sigma^n=$ id, so if there is an element of $Aut(\mathbb{F}\_q^\times)$ of order $n$, then you could just define exponentiation to $\omega$ (or any of its conjugates) to be that (group) automorphism of $\mathbb{F}\_q^\times$ (and set $0^\omega=0$). Note that the group of automorphisms $Aut(\mathbb{F}\_q^\times)\simeq Aut(\mathbb{Z}/(q-1)\mathbb{Z})$ is simply $(\mathbb{Z}/(q-1)\mathbb{Z})^\times$, which has order $\phi(q-1)$, so if $n\nmid\phi(q-1)$, this wouldn't work.


