Taking power of element of finite field?

Question

Let $\mathbb{F_{q^n}}$ be a finite field.

Assume $a , b \in \mathbb{F_{q^n}}$.

Is there any meaning to the power operation: $a^b$?

Answer 1

No, there is not. As some evidence that there is no natural way to define it, note that for instance, in $\mathbb{F}_3$, you would expect $2^{1+1+1}=2^1\cdot 2^1\cdot 2^1=2\cdot 2\cdot 2=2$ and also $2^{1+1+1}=2^0=1$.

In general, in a ring $R$, the exponentiation operation $a^n$ is defined if $a\in R$ and $n\in\mathbb{N}$ (not for $n\in R$): you can recursively define $a^0=1$ and $a^{n+1}=a\cdot a^n$. If $a$ is a unit, you can extend this to $n\in\mathbb{Z}$ by defining $a^n=b^{-n}$ when $n$ is negative, where $b$ is the multiplicative inverse of $a$.