I am trying to work out the multiplicative order of each non zero element in $F_7$.
Lets say I am looking at the number $3$. I know its order is $6$.
Instead of having to work out the powers of three and work out the remainder when divided by $7$, is their an easier way to calculate the order especially when the numbers get bigger?
The easiest way to find order of an element of $F_p$ by hand, is calculate remainder of powers of your number for all prime divisors of $p-1$, and then by use of them to calculate order of divisors of $p-1$ by the way that I explain below by help of an example:
For example you want to calculate order of $13 \in F_{31}$ :
$13*13=169 \equiv 14 \pmod {31}$
$13^3=13*13^2\equiv 13*14=182 \equiv 27 \pmod{31}$
Now $13^5=13^2*13^3 \equiv 14*27\equiv 6 \pmod {31}$
$13^{10}=13^5*13^5 \equiv 6*6 \equiv 5 \pmod {31}$
$13^{15}=13^5*13^{10} \equiv 6*5 = 30 \pmod {31}$
And so order of $13$ in $F_{31}$ is $30$.