Definitions

$⟨$ Definition A-3.1 $⟩$ Field Definitions

Ring: A set of elements which is an abelian group under the + operator, and closed, associative, and distributive on the ( $+,⋅$ ) operators.
Field: A set of elements which is an abelian group under both the $(+,⋅)$ operators (i.e., the set has an identity element and multiplicative inverses for all elements), and distributive on those operators.
Galois Field ( $GF (p^{n})$ ): A field with a finite number of elements (whose number must be $p^{n}$ for some prime $p$ and a positive integer $n$ ).
$ℤ_{p}$ ( $ℤ ∕ 𝑝ℤ$ ): The finite field of integer modulo $p$ , which is ${0, 1, 2, \dots . . . p - 1}$ where $p$ is a prime number. If $p$ is a prime number, $ℤ_{p}$ is always a finite field. This is also called a quotient ring of $p$ .