Complex powers in finite fields

There is no standard way to define this idea, and I doubt there is a useful one.

If I were trying to invent a definition, I'd say the following:

We should definitely have $(ab)^i = a^i b^i$ and $(a^i)^i=a^{-1}$. The first says that taking the $i$-th power is a homomorphism from the multiplicative group of the field to itself. Since the multiplicative group of a finite field is cyclic, the only homomorphisms from it to itself are of the form $a mapsto a^k$ for some integer $k$.

Which integers $k$ should we accept? The second property imposes $a^{k^2} = a^{-1}$ modulo $p$. This implies that $k^2 = -1 mod (p-1)$.

For most primes $p$, there are no solutions to $k^2 = -1 mod (p-1)$. In the cases that there are, you could define raising to the $i$ power to means raising to the $k$ power for such a $k$. For example, when $p=11$, you could define $a^i$ to mean $a^3$, and you would get the two properties above.

I wouldn't do this though. At least in the areas of math where I work, it is considered a bad idea to define an operation on $F_p$ that doesn't extend nicely to the fields $F_{p^j}$. And there will never be a good definition of raising to the $i$ power in $F_{p^2}$. In order to get one, you'd need a solution to $k^2 = -1 mod p^2-1$. But, for any prime other than $3$, $p^2-1$ is divisible by $3$, and $k^2+1$ is never divisible by $3$.

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