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

These are called the field axioms.

1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields
1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19$: Properties of $\Z_m$ as an algebraic system

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