if we have any positive integer m ,
the sum of a^m = -1 if (q-1)|m or 0 otherwise.
Note, the sum is over a belongs to Fq , where Fq is a field.
I appreciate your help,

if we have any positive integer m ,
the sum of a^m = -1 if (q-1)|m or 0 otherwise.
Note, the sum is over a belongs to Fq , where Fq is a field.
I appreciate your help,

Hint: 1)

2) .

But there exists (why?), so

, and since we have a group here the last sum runs over