Field

A field is a set $F$ together with two binary operations, written as $(x,y)\mapsto x+y$ [addition] and $(x,y)\mapsto x\cdot y$ [multiplication], such that the following axioms hold.

1. For all $a,b,c\in F$, one has $(a+b)+c=a+(b+c)$. [Associativity of addition]
2. There exists an element $0\in F$ such that $a+0=a$ for every $a\in F$. [Additive identity]
3. For every $a\in F$, there exists an element $-a\in F$ such that $a+(-a)=0$. [Additive inverse]
4. For all $a,b\in F$, one has $a+b=b+a$. [Commutativity of addition]
5. For all $a,b,c\in F$, one has $(ab)c=a(bc)$. [Associativity of multiplication]
6. There exists an element $1\in F$ with $1\neq 0$ such that $a1=a$ for every $a\in F$. [Multiplicative identity]
7. For every $a\in F$ with $a\neq 0$, there exists an element $a^{-1}\in F$ such that $aa^{-1}=1$. [Multiplicative inverse]
8. For all $a,b\in F$, one has $ab=ba$. [Commutativity of multiplication]
9. For all $a,b,c\in F$, one has $a(b+c)=ab+ac$. [Distributive property]