Field (mathematical concept) - Revision history

Rongzhou: /* Formal definition */

2026-10-16T15:12:53Z

Formal definition

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 Then $(F,+_F,\cdot_F)$ is a '''field'''.
 <blockquote>
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 <ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>

Rongzhou at 14:55, 16 October 2026

2026-10-16T14:55:06Z

← Older revision		Revision as of 14:55, 16 October 2026
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	== Formal definition ==		== Formal definition ==

	Let $F:\|F\| \ge 2$ be a set, and $+,\~~cdot~~$ two [[binary operation\|binary operations]] defined over $F$.		Let $F:\|F\| \ge 2$ be a set, and $+_F,\cdot_F$ two [[binary operation\|binary operations]] defined over $F$.

	If $\forall a,b,c \in F$:		If $\forall a,b,c \in F$:

	* $a+b=b+a;a \~~cdot~~ b = b \~~cdot~~ a$ ('''commutativity''')		* $a+_Fb=b+_Fa;a \cdot_F b = b \cdot_F a$ ('''commutativity''')
	* $(a+b)+c=a+(b+c);(a \~~cdot~~ b)\~~cdot~~ c = a \~~cdot~~ (b \~~cdot~~ c)$ ('''associativity''')		* $(a+_Fb)+_Fc=a+_F(b+_Fc);(a \cdot_F b)\cdot_F c = a \cdot_F (b \cdot_F c)$ ('''associativity''')
	* $\exists 0_F,1_F \in F: a+~~0_F~~=a, a \~~cdot~~ 1_F=a$ ('''identity''')		* $\exists 0_F,1_F \in F: a+_F0_F=a, a \cdot_F 1_F=a$ ('''identity''')
	* $\exists (-a) \in F: a + (-a)=0_F; \forall a \neq 0_F: \exists (a^{-1}) \in F, a\~~cdot~~ (a^{-1})=1_F$ ('''inverse''')		* $\exists (-a) \in F: a +_F (-a)=0_F; \forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot_F (a^{-1})=1_F$ ('''inverse''')
	* $a\~~cdot~~ (b+c)=a\~~cdot~~ b + a\~~cdot~~ c$ ('''distributivity''')		* $a\cdot_F (b+_Fc)=a\cdot_F b +_F a\cdot_F c$ ('''distributivity''')

	Then $(F,+,\~~cdot~~)$ is a '''field'''.		Then $(F,+_F,\cdot_F)$ is a '''field'''.

	'''Example''':		'''Example''':
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	\|-		\|-
	! $F$		! $F$
	! $(F,+,\~~cdot~~ )$ is field?		! $(F,+_F,\cdot_F )$ is field?
	\|-		\|-
	\| $\mathbb{N}$		\| $\mathbb{N}$
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	\| Yes		\| Yes
	\|}		\|}

			<blockquote>
			Just like in [[ring\|rings]], $+_F,\cdot_F$ do not have to take the usual definitions of $+,\cdot $.
			</blockquote>

			<blockquote>
			The smallest field is $(\{0_F,1_F\},+_F,\cdot_F)$
			</blockquote>

	<blockquote>		<blockquote>
	Alternatively, a field can be defined as a [[ring]] with additional properties:		Alternatively, a field can be defined as a [[ring]] with additional properties:

	* $a \~~cdot~~ b = b \~~cdot~~ a$ ('''multiplicative commutativity''')		* $a \cdot_F b = b \cdot_F a$ ('''multiplicative commutativity''')
	* $\exists 1_F \in F: a \~~cdot~~ 1_F=a$ ('''multiplicative identity''')		* $\exists 1_F \in F: a \cdot_F 1_F=a$ ('''multiplicative identity''')
	* $\forall a \neq 0_F: \exists (a^{-1}) \in F, a\~~cdot~~ (a^{-1})=1_F$ ('''multiplicative inverse''')		* $\forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot_F (a^{-1})=1_F$ ('''multiplicative inverse''')

	Also, note unlike for rings, there is no need to specify distributivity for each slot, since the slots for multiplication are symmetric in a field because of multiplicative commutativity.		Also, note unlike for rings, there is no need to specify distributivity for each slot, since the slots for multiplication are symmetric in a field because of multiplicative commutativity.
			</blockquote>

	<ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>		<ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>
	~~</blockquote>~~

	~~<blockquote>~~
	~~The smallest field is $\{0_F,1_F\}$~~
	~~</blockquote>~~

Rongzhou: /* Formal definition */

2026-10-16T14:21:37Z

Formal definition

← Older revision		Revision as of 14:21, 16 October 2026
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	<ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>		<ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>
			</blockquote>

			<blockquote>
			The smallest field is $\{0_F,1_F\}$
	</blockquote>		</blockquote>

Rongzhou at 14:03, 16 October 2026

2026-10-16T14:03:43Z

← Older revision		Revision as of 14:03, 16 October 2026
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	\|}		\|}

	<ref>{{#cite:Q2202}}</ref>		<blockquote>
			Alternatively, a field can be defined as a [[ring]] with additional properties:

			* $a \cdot b = b \cdot a$ ('''multiplicative commutativity''')
			* $\exists 1_F \in F: a \cdot 1_F=a$ ('''multiplicative identity''')
			* $\forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot (a^{-1})=1_F$ ('''multiplicative inverse''')

			Also, note unlike for rings, there is no need to specify distributivity for each slot, since the slots for multiplication are symmetric in a field because of multiplicative commutativity.

			<ref>{{#cite:Q2202}}</ref><ref>{{#cite:Q2204}}</ref>
			</blockquote>

Rongzhou: Created page with "A '''field''' is a set closed under two binary operations, with some special properties by definition. == Formal definition == Let $F:|F| \ge 2$ be a set, and $+,\cdot$ two binary operations defined over $F$. If $\forall a,b,c \in F$: * $a+b=b+a;a \cdot b = b \cdot a$ ('''commutativity''') * $(a+b)+c=a+(b+c);(a \cdot b)\cdot c = a \cdot (b \cdot c)$ ('''associativity''') * $\exists 0_F,1_F \in F: a+0_F=a, a \cdot 1_F=a$ ('''identity''') * $\e..."

2026-10-16T13:10:02Z

Created page with "A '''field''' is a set closed under two binary operations, with some special properties by definition. == Formal definition == Let $F:|F| \ge 2$ be a set, and $+,\cdot$ two binary operations defined over $F$. If $\forall a,b,c \in F$: * $a+b=b+a;a \cdot b = b \cdot a$ ('''commutativity''') * $(a+b)+c=a+(b+c);(a \cdot b)\cdot c = a \cdot (b \cdot c)$ ('''associativity''') * $\exists 0_F,1_F \in F: a+0_F=a, a \cdot 1_F=a$ ('''identity''') * $\e..."

New page

A '''field''' is a [[set]] closed under two binary operations, with some special properties by definition.

== Formal definition ==

Let $F:|F| \ge 2$ be a set, and $+,\cdot$ two [[binary operation|binary operations]] defined over $F$.

If $\forall a,b,c \in F$:

* $a+b=b+a;a \cdot b = b \cdot a$ ('''commutativity''')
* $(a+b)+c=a+(b+c);(a \cdot b)\cdot c = a \cdot (b \cdot c)$ ('''associativity''')
* $\exists 0_F,1_F \in F: a+0_F=a, a \cdot 1_F=a$ ('''identity''')
* $\exists (-a) \in F: a + (-a)=0_F; \forall a \neq 0_F: \exists (a^{-1}) \in F, a\cdot (a^{-1})=1_F$ ('''inverse''')
* $a\cdot (b+c)=a\cdot b + a\cdot c$ ('''distributivity''')

Then $(F,+,\cdot)$ is a '''field'''.

'''Example''':

{| class="wikitable"
|-
! $F$
! $(F,+,\cdot )$ is field?
|-
| $\mathbb{N}$
| No, $a=1 \in \mathbb{N}: \nexists (-1) \in \mathbb{N}$
|-
| $\mathbb{Z}$
| No, $a=2 \in \mathbb{N}: \nexists (2^{-1}) \in \mathbb{Z}$
|-
| $\mathbb{Q}$
| Yes
|-
| $\mathbb{R}$
| Yes
|}

<ref>{{#cite:Q2202}}</ref>