Consider the following problem:
If |K| = q and (n ,q) = 1 and F is a splitting field of $x^n-1_K$ over K, then prove that k = [F : K] is the least positive integer k such that $n | (q^k - 1)$.
Attempt : $|F| = q^k$ . Let p be a root of $x^n -1$=0 . But I am unable to think of any other details on how the question should be approached and I need help.
So, Can you please outline a solution.
Thank you.
Hint: The roots to $x^n=1$ form a multiplicative group. Under what hypothesis does this group fit within $F^*$?