Definitions

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

Ring: A set $R$ that is an abelian group under addition $(+)$ , equipped with a multiplication $(⋅)$ that is closed and associative, and such that multiplication distributes over addition on both sides: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in R$ . (Multiplication is not necessarily commutative (e.g., a matrix multiplication), and an identity element for $(⋅)$ is optional unless stated “ring with unity”.)
Field: A set $F$ that is an abelian group under $(+)$ , whose nonzero elements $F^{\times} = F ∖ {0}$ form an abelian group under $(⋅)$ , with multiplication distributing over addition.
Galois Field ( $GF (p^{n})$ ): A field with a finite number of elements, necessarily $p^{n}$ for some prime $p$ and positive integer $n$ .
$ℤ_{p}$ ( $ℤ ∕ 𝑝ℤ$ ): For a prime $p$ , the set ${0, 1, \dots, p - 1}$ with addition and multiplication modulo $p$ forms a finite field. More generally, for any integer $m \geq 2$ , $ℤ_{m}$ is a commutative ring, and it is a field iff $m$ is prime.