Non-trivial 93rd roots of unity in finite fields

Question

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Finding the values of $n$ for which $\mathbb{F}_{5^{n}}$, the finite field with $5^{n}$ elements, contains a non-trivial $93$rd root of unity

For which of the following value of $n$, does the finite field $\Bbb F$, with $5^n$ elements contain a non trivial $93$-rd root of unity?

1. 92
2. 30
3. 15
4. 6

Answer 1

The multiplicative group of a finite field $\Bbb F$ with $\mid \Bbb F\mid=q$ is cyclic and so, by the theory of cyclic groups, it contains a unique subgroup of order $d$ for each divisor $d$ of $q-1$. Thus, the question becomes: for which values of $n$ we have that $93$ divides $5^n-1$?

Now:

• $5^{92}\equiv67^{23}\equiv40\bmod93$,

• $5^{30}\equiv(5^6)^5\equiv1\bmod93$,

• $5^{15}\equiv56^3=32\not\equiv1\bmod93$,

• $5^6\equiv1\bmod93$.

So, of the $n$ listed, the answer is 6 and 30.