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Suppose that $g$ is a primitive root modulo $p$. Show that, modulo $p^h$ for $h\geq 2$, every primitive root has the form $g'=g+np$ for a certain integer $n$. The proof of the previous statement should appear in every book of elementary number theory but, in this moment, I don't have any book with me and I should prove this.

My attempt: I believe one possible way to prove it is to show that $g'\equiv g$ mod $p$, that is every primitive root modulo $p^h$ is also a primitive root modulo $p$, but I'm stuck on how to prove this simple statement.

Thanks in advance for any help.

P.S.

Of course I'd appreciate every possible proof...I believe that this can be proved in several ways.

PITTALUGA
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    See http://math.stackexchange.com/questions/170648/prove-that-any-primitive-root-r-of-pn-is-also-a-primitive-root-of-p – lab bhattacharjee May 17 '16 at 08:48
  • Thanks a lot @labbhattacharjee – PITTALUGA May 17 '16 at 09:04
  • @PITTALUGA sorry i don't get it, are these two post the same question? your question is about from primitive root $g$ of $p$ generating a primitive root in the form of $g+np$ for $p^h$; while the one May linked is about any primitive root of $p^n$ is also a primitive root of $p$... – athos Sep 20 '20 at 10:22

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