A [[Field]] is a [[Set|set]] equipped with an [[Algebraic Structure|algebra]] consisting of multiplication, division, addition, and subtraction. $ \huge \begin{align} a+(b+c) &= (a+b)+c \\ a\cdot(b \cdot c) &= (a \cdot b)\cdot c \\ a+b &= b + a \\ a \cdot b &= b \cdot a \\ \forall a: a+0 &= 0 , 0 \in F \\ \forall a: a \cdot 1 &= 1 , 1 \in F \\ \forall a \exists (-a) &: a+(-a) = 0 \\ \forall a\in F\setminus\set{0}\exists a^{-1} &: a \cdot a^{-1} = 1\\ a \cdot (b+c) &= a \cdot b + a \cdot c \end{align} $