Definition (Field)

A field is a triple $(\mathbb{F},+,\cdot)$ such that

1. $(\mathbb{F},+)$ is an abelian group
2. $(\mathbb{F}^{*},\cdot)$ (where $\mathbb{F}^{*}=\mathbb{F}\setminus\{ 0 \}$) is an abelian group
3. Distributivity: $$\begin{align*} a\cdot(b+c)=ab+ac\\ (b+c)\cdot a=ba+ca \end{align*}$$

Definition (Finite field)

$\mathbb{F}_{n}$​ denotes a finite field with $n$ elements.