In , and . In , you have and . So, your multiplication table for the extension of by is misleading since . So, you have:
Since there are only three non-zero elements, is indeed the generator of that cyclic group. Next, for the field of 9 elements, you need to extend by . There will be 8 nonzero elements of that field extension, and they will, indeed, form a cyclic group. Just make sure you remember how addition and multiplication work in the finite fields. It works just like modular arithmetic.