How to find multiplicative orders of all elements in field $\Bbb F$ (say $\Bbb F_{13}$)? 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 $\Bbb F$?

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 $2$.

