Theorems

$⟨$ Theorem A-3.3 $⟩$ Field Theorems

1.: Size of Finite Field: A finite field is called Galois field and always has $p^{n}$ elements (where $p$ is a prime and $n$ is a positive integer).
2.: Isomorphic Fields: Any two finite fields, $𝔽_{1}$ and $𝔽_{2}$ with the same number of elements are isomorphic (i.e., there exists a bi-jective one-to-one mapping function $f : 𝔽_{1} \to 𝔽_{2}$ and the algebraic operations $(+,⋅)$ preserve correctness among newly mapped elements). In other words, there exists a mapping function $f : 𝔽_{1} \to 𝔽_{2}$ comprised of the field operators ( $+$ , $\cdot$ ). For such an isomorphic function $f$ , for any $a, b \in 𝔽_{1}$ , $f (a + b) = f (a) + f (b)$ and $f (a \cdot b) = f (a) \cdot f (b)$