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I am learning about groups generators and there are questions regarding finding the smallest positive integer that generates $F_p^* $. I am currently trying to calculate based on each element of the set, but I wonder if there is a more systematic way to do this kind of question? For example, $F_{17}^*$ would be big enough for me to feel calculating each element by hands not efficient anymore!

ajinomoto
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    Another wording of this question is "what is the smallest primitive root modulo $p$?", which might help you search for what is known. There's no formula, and indeed I don't think we can do that much better than try each integer in turn (although there are fast ways to "try" integers). – Greg Martin Feb 05 '23 at 22:57
  • What is $F^*_p$? – Shaun Feb 05 '23 at 23:06
  • In the note, I see they define $F_p^* $ as the set of number from 1 to $p-1$ that are relatively prime to $p$. Do you know if there is any formula for determine the number of generators $F_p^* $ have if $p$ is prime? – ajinomoto Feb 05 '23 at 23:11
  • The multiplicative group of the finite field $\mathbb F_p^\times$ is cyclic of order $p-1$, so the number of generators is the Euler totient function $\phi(p-1)$ – J. W. Tanner Feb 05 '23 at 23:37

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