There is the following problem from my text book. I just wanna know if I understand the solution correctly.
For which of the following values of n does the finite field F5n with 5n elements contain a non trivial 93 rd root of unity? 1. 92 2. 30 3. 15 4. 6
The theorem I used was as follows
"Let F be a field and H , the group of nth roots of unity in F, be a subgroup of the multiplicative group FX. Then H is cyclic of some order m such that m divides n. If, in addition, F is finite with order q, |H| = (n, q-1)"
First up, the multiplicative group FX has |F| -1 elements. Since H is a subgroup of FX , o(H) should divide |F| -1. Since H is cyclic, there is a generator whose order is the same as the order of H. So if 'a' is a generator of H, o(a) should divide |F| -1.
Since H is a group of the nth roots of unity in F, the nth power of every element is 1, the identity element with respect to H and FX . So, a93 =1. This means 93 should divide |F| -1.
To sum it up, 93 should divide 5n - 1.
Have I been right so far?
Can someone help me find the right option from here on?