Clarification regarding understanding a solution - Finite field

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 F_{5n} with 5^{n} 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 F**^{X}. 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 **F**^{X }has |F| -1 elements. Since H is a subgroup of **F**^{X }, 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 **F**^{X }. So, a^{93} =1. This means 93 should divide |F| -1.

To sum it up, 93 should divide 5^{n }- 1.

Have I been right so far?

Can someone help me find the right option from here on?

Thanks,

Re: Clarification regarding understanding a solution - Finite field

well, not quite.

suppose (for example) that F contains all 3rd roots of unity, but no primitive 9th root of unity. well all 3rd roots of unity are also 9th roots of unity, if:

a^{3} = 1 (mod q-1) then:

a^{6} = (a^{3})^{3} = 1^{3} = 1 (mod q-1).

but the subgroup of 9-th roots of unity in F doesn't have order 9, but order 3.

for example F_{25} has a cyclic multiplicative group of order 24, which therefore has a cyclic subgroup of order 3, but no cyclic subgroup of order 9 since 9 does not divide 24

(if the generator of F* is b, then b^{8} is a cube root of 1, so b^{8} is a 9-th root of 1 as well:

(b^{8})^{9} = b^{72} = (b^{24})^{3} = 1^{3} = 1).

so all you can say is 93 and 5^{n}-1 have a common factor > 1. this common factor may be 3,31 or 93.

if F_{5n} is supposed to contain ALL 93-rd roots of unity, THEN 93 would have to divide 5^{n}-1.

Re: Clarification regarding understanding a solution - Finite field

am able to get your point Deveno... i could find the incorrect route i took.

but can you help me move ahead with this problem?