# Non-trivial 93rd roots of unity in finite fields [duplicate]

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
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## marked as duplicate by Dilip Sarwate, Henry T. Horton, Old John, Micah, rschwiebDec 19 '12 at 1:31

I totally forgot that this is a dup. Good job spotting that, guys. –  Jyrki Lahtonen Dec 19 '12 at 8:09

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.

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What's with the intermediate steps? Why those particular factorisations? –  Ben Millwood Dec 18 '12 at 14:00
thanks for ur precious advice. –  Alka Goyal Dec 18 '12 at 16:15
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