How to find multiplicative orders of all elements in F 13?

Question

How to find multiplicative orders of all elements in F 13?

I am working on some Finite fields and I was referring to some online class material. Is there any way to find this?

Answer 1

The non-zero elements in F 13 form a multiplicative group of order 12. You can represent them by the numbers 1, 2, 3, ..., 12. Algebra tells you that the group is cyclic. It turns out that 2 is a generator. Knowing the order of an element g in a group G it is straight forward to determine the order of any element on the form g^i. You can use this to determine the orders of all the elements.

A different method is to directly use the definition of the order of an element. That is for each element you calculate g, g^2, g^3, g^4, ... The smallest number d for which g^d = 1 is the order of that element. Given the small size of the group F 13* this is quite doable.