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: $$\begin{align*} a\cdot(b+c)=ab+ac\\ (b+c)\cdot a=ba+ca \end{align*}$$