## FANDOM

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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:

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 $a+(-a)=0=(-a)+a$ $a\cdot (a^{-1})=1=(a^{-1})\cdot a, \text{ if } a\ne 0$