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I am working on some finite fields and I was referring to some online class material. Is there any way to find the multiplicative orders of all elements in a field $\Bbb F$?

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up vote 3 down vote accepted

Just compute the powers: \begin{align} 2^2&\equiv_{13}4 &2^3&\equiv_{13}8 & 2^4=4^2&\equiv_{13}16\equiv_{13}3& 2^5&\equiv_{13}6&2^6&\equiv_{13}12\equiv_{13}-1\\ 3^2&\equiv_{13}9\equiv_{13}-4 &3^3&\equiv_{13}-12\equiv_{13}1\\ 5^2&\equiv_{13}-1\\ 6^2&\equiv_{13}-3&6^3&\equiv_{13}-5&6^4&\equiv_{13}-4&6^5&\equiv_{13}2&6^6&\equiv_{13}-1 \end{align} and $7\equiv_{13}-6$ etc. It follows that

  • $2,6,7,11$ have order $12$,
  • $3,9$ have order $3$,
  • $4,10$ have order $6$,
  • $5,8$ have order $4$,
  • $12$ has order $2$.
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