I'm studying cyclic codes. I read the following theorem :
$\mathbb{F}_q$ is a finite field with q elements. Let's consider $X^n - 1 \in \mathbb{F}_q[X]$.
If $GCD(q,n) = 1 $,
Then $\exists \alpha \in \overline{\mathbb{F}_q}$ such that $X^n - 1 = \prod_{i=0}^{n-1}(X-\alpha^i)$
But, what is $\overline{\mathbb{F}_q}$ ? I didn't find it anywhere else...
Whatever you read the symbol $\overline{k}$ should be defined somewhere. There are many different types of "closures".
Anyway in the field theory (and it seems to agree with the context of your question) it typically means the algebraic closure of $k$, i.e. the "smallest" field $L$ containing a copy of $k$ as its subfield and such that every polynomial of positive degree over $L$ has a root in $L$.
These algebraic closures are known to always exist and are unique for $k$.
Note that the algebraic closure of any (finite or not) field is infinite.