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Im working on this Exercise

I can do do part b) but Im stuck on part c).

I know that if $e$ is a positive factor of $p-1$ then the equation: $$X^e \equiv 1 \quad \textrm{mod p} $$ has exactly $e$ solutions modulo $p$, the proof of which involves Fermat's Little Theorem and Lagrange's Theorem so I think it may be relevant but I don't know how to apply it.

Any suggestions?

Connor Bishop
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  • Do you know about primitive roots? – Emre May 19 '16 at 10:10
  • One way to proceed is to observe that $f:\Bbb{Z}_p^\to\Bbb{Z}_p^, f(x)=x^{(p-1)/d}$, is a homomorphism of groups. You seem to know the size of the kernel of this homomorphism. Are you familiar with results relating the size of the image to the sizes of the domain and the kernel? – Jyrki Lahtonen May 19 '16 at 10:12
  • See also : http://math.stackexchange.com/questions/871817/congruence-of-nth-degree and http://math.stackexchange.com/questions/806684/xn-2mod-13 – lab bhattacharjee May 19 '16 at 10:16

1 Answers1

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Each non-zero $x^{(p-1)/d}$ is a root of $x^d\equiv 1\pmod{p}$. So, there are at most $d$ distinct values of $x^{(p-1)/d}$.

Emre
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