Almost correct, except for the fictional part about 3 being an N-th root of unity in GF(7) for all N. In reality 3 is a primitive element of the field - i.e. a

Almost correct, except for the fictional part about 3 being an N-th root of unity in GF(7) for all N. In reality 3 is a primitive element of the field - i.e. a generator of the multiplicative group, aka a primitive 6th root of unity. So it's confusing the concepts "primitive element" and "primitive N-th root".