Field

Definition

A field $F$ is a set on which two operations $+$ (addition) and $\cdot$ (multiplication) are defined so that the followings hold.

For any $x,y,z\in F$,

(A1) $x+y=y+x$ 　(Commutativity of addition)

(A2) $(x+y)+z=x+(y+z)$ 　(Associativity of addition)

(A3) $\exists !0\in F$ such that $\forall x\in F:x+0=x$ 　(Existence of additive identity)

(A4) $\forall x\in F:\exists !(-x)\in F$ such that $x+(-x)=0$ 　(Existence of additive inverse)

(M1) $xy=yx$ 　(Commutativity of multiplication)

(M2) $(xy)z=x(yz)$ 　(Associativity of multiplication)

(M3) $\exists !1\in F$ such that $\forall x\in F:1x=x$ 　(Existence of multiplicative identity)

(M4) $\forall x\in F\setminus \{0\}:\exists !x^{-1}\in F$ such that $x{x}^{-1}=1$ 　(Existence of multiplicative inverse)

(D) $x(y+z)=xy+xz$ 　(Distributivity of multiplication over addition)

Examples

• The set of rational numbers $(\mathbb{Q})$, real numbers $(\mathbb{R})$, and complex numbers $(\mathbb{C})$ are fields.
• The set $B=\{0,1\}$ with boolean addition / multiplication is a field.