Definition Field
A \textit{field} is a triple $(K, +, \cdot)$ consisting of a set $K$ and two internal operations $+$ (addition) and $\cdot$ (multiplication), such that:
K1 | $(K, +)$ is an abelian group. Its neutral element is denoted by $0$. |
K2 | $(K^\times, \cdot)$ is an abelian group, where $K^\times = K \setminus \{0\}$. Its neutral element is denoted by $1$. In particular, for $a, b \in K^\times$, $a \cdot b \in K^\times$. |
K3 | The distributive laws hold: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ and $(b + c) \cdot a = (b \cdot a) + (c \cdot a)$ for all $a, b, c \in K$. |