Definition:Field Axioms

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Definition

The properties of a field are as follows.

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

\((A0)\)   $:$   Closure under addition      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x + y \in F \)             
\((A1)\)   $:$   Associativity of addition      \(\displaystyle \forall x, y, z \in F:\) \(\displaystyle \paren {x + y} + z = x + \paren {y + z} \)             
\((A2)\)   $:$   Commutativity of addition      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x + y = y + x \)             
\((A3)\)   $:$   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
\((A4)\)   $:$   Inverse elements for addition      \(\displaystyle \forall x: \exists x' \in F:\) \(\displaystyle x + x' = 0_F = x' + x \)             $x'$ is called a negative element
\((M0)\)   $:$   Closure under product      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x \circ y \in F \)             
\((M1)\)   $:$   Associativity of product      \(\displaystyle \forall x, y, z \in F:\) \(\displaystyle \paren {x \circ y} \circ z = x \circ \paren {y \circ z} \)             
\((M2)\)   $:$   Commutativity of product      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x \circ y = y \circ x \)             
\((M3)\)   $:$   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
\((M4)\)   $:$   Inverse elements for product      \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x \)             
\((D)\)   $:$   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.


Sources