Regarding "hidden work" of MAGMA Order function

Question

The computational algebra system MAGMA has a function called Order. What it does is find the multiplicative order of its argument, which must be a nonzero element of a field. This question is asking about how this order is calculated. Can anyone point me to a reference explaining that or explain that to me in an answer?

Answer 1

I'm guessing that in most cases, it knows how to represent your element $\beta = \alpha^k$, where $\alpha$ is a generator of the field. Then it computes the order by considering $(q-1)/GCD(k, q-1)$.