Number of solutions of $x^n = 1 \pmod{p}$ for every integer $n$

Asked Nov 08 '17 at 09:06

Active Nov 08 '17 at 10:48

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In the integers modulo $p$, $p$ a prime, there are at most $n$ solutions of $x^n = 1 \pmod{p}$ for every integer $n$.

I was wondering if there is any proof which only uses group theory and no field theory. This is a question asked in supplementary exercise of Herstein's group theory.

edited Nov 08 '17 at 09:16

asked Nov 08 '17 at 09:06

Shak

1,214

Maybe you can argue about the order of x? – TheOscillator Nov 08 '17 at 09:19
Without a bound on $x$ this is trivially false, e. g. $x = 1$, $x = 3$ for $n = 1$, $p = 2$. – orlp Nov 08 '17 at 10:12
Maybe you find this post useful. – Amin235 Nov 08 '17 at 11:43
Looking at Herstein, we see that the next exercise is: "Prove that the nonzero integers mod $p$ under multiplication form a cyclic group". So, this result cannot be used to answer the question. – lhf Nov 08 '17 at 23:23

Number of solutions of $x^n = 1 \pmod{p}$ for every integer $n$

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