In A Course in Arithmetic (Serre) he beings by proving the quadratic reciprocity law (Thereom $6$) as follows:
Let $p$ and $l$ be two distinct primes different from $2$. Let $\Omega$ be an algebraic closure of $\mathbf{F}_{p}$ and let $w \in \Omega$ be a primitive $l$-th root of unity. If $x \in \mathbf{F}_{l}$, the element $w^{x}$ is well defined since $w^{l} = 1$...
My question is as follows: I don't understand why $w^{x}$ is defined at all. I know that $w^{n}$ is defined if $n$ is an integer but $x$ is an element of the field $\mathbf{F}_{l}$. So what do we even mean by $w^{x}$ in a precise sense? Does it still reside in $\Omega$? I can't think of a polynomial where one of its roots is $w^{x}$.
The field $\mathbf{F}_l$ is just $\mathbb{Z}/l\mathbb{Z}$, so an element of it is an equivalence class of integers modulo $l$. Since $w^l=1$, you can raise $w$ to the power of any representative of the equivalence class and the outcome will not depend on the choice of representative. In other words, $w^x$ is defined as $w^n$, where $n$ is any integer in the equivalence class $x$.