We know that the binary splitting field of the equation $x^n-1$ is the $GF(2^q)$ where $q$ is the least positive number such that $n\mid 2^q-1$.
My question: what is the binary splitting field of the equation $x^n-\alpha$ such that $\alpha$ is the element of $GF(2^k)$.
In fact, I want to ask: Is there a relation between the binary splitting field of the equations $x^n-1$ and $x^n-\alpha$ such that $\alpha \neq 1$.
Edit(1): Edit question by comment of @ ancient mathematician.