Theorems

$⟨$ Theorem A-3.3 $⟩$ Field Theorems

1.: Size of Finite Field: Every finite field (also called a Galois Field) has $p^{n}$ elements for some prime $p$ and positive integer $n$ , conversely, for each such prime power $p^{n}$ , there exists a finite field of order $p^{n}$ (unique up to isomorphism).
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 bijection $f : 𝔽_{1} \to 𝔽_{2}$ such that for all $a, b \in 𝔽_{1}$ , $f (a + b) = f (a) + f (b)$ and $f (𝑎𝑏) = f (a) f (b)$ .