# Definition:Field Axioms

## Definition

The properties of a field are as follows.

For a given field $\struct {F, +, \circ}$, these statements hold true:

 $(\text A 0)$ $:$ Closure under addition $\displaystyle \forall x, y \in F:$ $\displaystyle x + y \in F$ $(\text A 1)$ $:$ Associativity of addition $\displaystyle \forall x, y, z \in F:$ $\displaystyle \paren {x + y} + z = x + \paren {y + z}$ $(\text A 2)$ $:$ Commutativity of addition $\displaystyle \forall x, y \in F:$ $\displaystyle x + y = y + x$ $(\text A 3)$ $:$ Identity element for addition $\displaystyle \exists 0_F \in F: \forall x \in F:$ $\displaystyle x + 0_F = x = 0_F + x$ $0_F$ is called the zero $(\text A 4)$ $:$ Inverse elements for addition $\displaystyle \forall x: \exists x' \in F:$ $\displaystyle x + x' = 0_F = x' + x$ $x'$ is called a negative element $(\text M 0)$ $:$ Closure under product $\displaystyle \forall x, y \in F:$ $\displaystyle x \circ y \in F$ $(\text M 1)$ $:$ Associativity of product $\displaystyle \forall x, y, z \in F:$ $\displaystyle \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$ $(\text M 2)$ $:$ Commutativity of product $\displaystyle \forall x, y \in F:$ $\displaystyle x \circ y = y \circ x$ $(\text M 3)$ $:$ Identity element for product $\displaystyle \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:$ $\displaystyle x \circ 1_F = x = 1_F \circ x$ $1_F$ is called the unity $(\text M 4)$ $:$ Inverse elements for product $\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:$ $\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x$ $(\text D)$ $:$ Product is distributive over addition $\displaystyle \forall x, y, z \in F:$ $\displaystyle x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z}$

These are called the field axioms.