Definition (Ring of Polynomials of Order $p$)

We define the ring of polynomials of order $p$ where $p$ is prime in terms of the Finite Field of Order p, $\mathbb{F}_{p}$ and we denote it as $$\mathbb{F}_{p}[x]$$where elements are of the form $$a_{n}x^{n}+\dots+a_{1}x+a_{0}$$ where each $a_{i}\in\mathbb{F}_{p}$.

Definition (The Set of Residue Classes mod p)

The set of residue classes $\pmod{p}$ is defined as $$\mathbb{Z}/p\mathbb{Z}=\{ [1],[2],\dots,[p-1] \}$$

Definition (Finite Field of Order p)

If $p$ is a prime, $(\mathbb{Z}/p\mathbb{Z},+,\cdot)$ () is a Finite field of $p$ elements. We usually denote this as $$\mathbb{F}_{p}$$