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

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