FANDOM


A field is a set of numbers together with the Field axioms.

Let F be a field. Then, assuming a, b, and c are elements of F, the following axioms hold:

name addition multiplication
closure $ a+b \in F $ $ a\cdot b \in F $
commutative $ a+b=b+a $ $ a\cdot b=b\cdot a $
associative $ a+(b+c)=(a+b)+c $ $ a\cdot (b\cdot c)=(a\cdot b)\cdot c $
distributive $ a\cdot (b+c)=a\cdot b+a\cdot c $
identity $ a+0=0+a=a $ $ (a\cdot 1)=(1\cdot a)= a $
inverse $ a+(-a)=0=(-a)+a $ $ a\cdot (a^{-1})=1=(a^{-1})\cdot a, \text{ if } a\ne 0 $