A Sequence Over a Field

You are given the set $\mathbb{Z}_p = \{0,1, \ldots p-1\}$ where $p$ is an odd prime. We define the operation $\oplus$ on the elements of this set as $a \oplus b= (a+b) % p$. We also define the operation $\otimes$ on the elements of this set as $a \otimes b = (a \times b) %p$. Now, we notice that $(\mathbb{Z}_p,\oplus,\otimes)$ form a field. Let us refer to the field as $\mathbb{F}_p$. Now, we co