Equivalence of Definitions of Field (Abstract Algebra)

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Theorem

The following definitions of the concept of Field (Abstract Algebra) are equivalent:

Definition $1$

$\struct {F, +, \times}$ is a field if and only if:

$(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group

$(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set {0_F}$

$(3): \quad$ the operation $\times$ distributes over $+$.

Definition $2$

Let $\struct {F, +, \times}$ be an integral domain such that every non-zero element $a$ of $F$ has a multiplicative inverse $a^{-1}$ such that:

$a \times a^{-1} = 1_F = a^{-1} \times a$

where $1_F$ denotes the unity of $\struct {F, +, \times}$.

Then $\struct {F, +, \times}$ is a field.

Definition $3$

Let $\struct {F, +, \times}$ be a commutative ring with unity $\struct {F, +, \times}$ such that every non-zero element $a$ of $F$ has a multiplicative inverse:

$a^{-1}$ such that $a \times a^{-1} = 1_F = a^{-1} \times a$

where $1_F$ denotes the unity of $\struct {F, +, \times}$.

Then $\struct {F, +, \times}$ is a field.

Definition $4$

Let $\struct {F, +, \times}$ be a non-trivial division ring whose ring product is commutative.

Then $\struct {F, +, \times}$ is a field.

Definition $5$

$\struct {F, +, \times}$ is a field if and only if it fulfils the conditions of the field axioms, as follows:

$(\text A 0)$	$:$	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 \times y \in F $
$(\text M 1)$	$:$	Associativity of product	$\ds \forall x, y, z \in F:$	$\ds \paren {x \times y} \times z = x \times \paren {y \times z} $
$(\text M 2)$	$:$	Commutativity of product	$\ds \forall x, y \in F:$	$\ds x \times y = y \times 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 \times 1_F = x = 1_F \times 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 \times x^{-1} = 1_F = x^{-1} \times x $
$(\text D)$	$:$	Product is distributive over addition	$\ds \forall x, y, z \in F:$	$\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} $

These are called the field axioms.

Proof

Definition $(1)$ implies Definition $(5)$

Let $\struct {F, +, \times}$ be a field by definition $1$.

Recall definition $1$ of Field (Abstract Algebra):

$\struct {F, +, \times}$ is a field if and only if:

$(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group

$(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set {0_F}$

$(3): \quad$ the operation $\times$ distributes over $+$.

By the definition of abelian group:

$(\text A 0): \quad \struct {F, +}$ is closed

$(\text A 1): \quad +$ is associative on $F$

$(\text A 2): \quad +$ is commutative on $F$

$(\text A 3): \quad$ there exists an identity element $0_F$ for $+$

$(\text A 4): \quad$ for every $a \in F$ there exists an inverse element $-a$ for $+$

Then:

$(\text M 0): \quad \struct {F, \times}$ is closed under addition

$(\text M 1): \quad \times$ is associative on $F$

$(\text M 2): \quad \times$ is commutative on $F$

$(\text M 3): \quad$ there exists an identity element $1_F$ for $\times$

$(\text M 4): \quad$ for every $a \in F$ there exists an inverse element $a^{-1}$ for $\times$

Finally we are given:

$(\text D): \quad \times$ is distributive over $+$ in $\struct {F, +, \times}$.

Thus $\struct {F, +, \times}$ is a field by definition $5$.

$\Box$

Definition $(5)$ implies Definition $(1)$

Let $\struct {F, +, \times}$ be a field by definition $5$.

Recall definition $5$ of Field (Abstract Algebra):

$\struct {F, +, \times}$ is a field if and only if it fulfils the conditions of the field axioms, as follows:

$(\text A 0)$	$:$	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 \times y \in F $
$(\text M 1)$	$:$	Associativity of product	$\ds \forall x, y, z \in F:$	$\ds \paren {x \times y} \times z = x \times \paren {y \times z} $
$(\text M 2)$	$:$	Commutativity of product	$\ds \forall x, y \in F:$	$\ds x \times y = y \times 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 \times 1_F = x = 1_F \times 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 \times x^{-1} = 1_F = x^{-1} \times x $
$(\text D)$	$:$	Product is distributive over addition	$\ds \forall x, y, z \in F:$	$\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} $

These are called the field axioms.

From $(\text A 0)$ to $(\text A 4)$, $\struct {F, +}$ is an abelian group.

From $(\text M 0)$ to $(\text M 4)$, $\struct {F \setminus \set 0_F, \times}$ is an abelian group.

Finally from $D$, $\times$ is distributive over $+$.

Thus $\struct {F, +, \times}$ is a field by definition $1$.

$\Box$

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

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