Field axioms

Field axioms: Let $F$ be a set equipped with binary operations $+$ and $\cdot$ and distinguished elements $0,1\in F$ with $0\neq 1$. The following are required for all $a,b,c\in F$:

• (Additive associativity) $(a+b)+c=a+(b+c)$.
• (Additive commutativity) $a+b=b+a$.
• (Additive identity) $a+0=a$.
• (Additive inverse) there exists $-a\in F$ such that $a+(-a)=0$.
• (Multiplicative associativity) $(ab)c=a(bc)$.
• (Multiplicative commutativity) $ab=ba$.
• (Multiplicative identity) $a1=a$.
• (Multiplicative inverse) if $a\neq 0$ then there exists $a^{-1}\in F$ such that $aa^{-1}=1$.
• (Distributivity) $a(b+c)=ab+ac$.

These axioms say exactly that $F$ is a field (equivalently, a commutative ring in which every nonzero element is invertible). Together with the order axioms and the completeness axiom , they characterize the real number system up to isomorphism.