Field

set with 2 binary operations, usually referred to as addition $+$ and multiplication $\times$, with the properties:

$$\{\begin{array}{llc}a+(b+c)=(a+b)+c& \text{additive associativity}& \\ a\times (b\times c)=(a\times b)\times c& \text{multiplicative associativity}& \\ a+b=b+a& \text{additive commutativity}& \\ a\times b=b\times a& \text{multiplicative commutativity}& \\ \exists 0:\forall a,a+0=a& \text{additive identity}& \\ \exists 1:\forall a,a\times 1=a& \text{multiplicative identity}& \\ \exists (-a)\forall a\ne 0:a+(-a)=0& \text{additive inverse}& \\ \exists a^{-1}\forall a\ne 0:a{a}^{-1}=1& \text{multiplicative inverse}& \\ a\times (b+c)=a\times b+a\times c& \text{distributivity}& \end{array}$$