Field
Concept history
A revision trail for this concept page.
Revision 780
8/1/2026, 9:33:43 AM · Sequoia
Concept edited
Recorded titleField
Recorded typeDefinition
Compare with revision 7706 changed lines
1
A field $(\mathbb{K},+,\times)$ is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]] with respect to the multiplicative law $\times$.1
A division ring $(R,+,\times)$ is a [[Ring|ring]] such that every nonzero element is [[invertible|invertible]] with respect to the multiplicative law $\times$. 2
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If the law $\times$ is also [[commutative|commutative]], then $R$ is called a field.4
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##### Remarks4
- Some authors (notably in the French literature) do not require a field to be commutative. The corresponding concept in English is called a [[division ring|division ring]].6
- Some authors (notably in the French literature) do not require a field to be [[commutative|commutative]]. Then they call "field" a division ring.Revision 770
7/31/2026, 3:25:30 PM · Sequoia
Concept edited
Compare with revision 7632 changed lines
1
A field is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]].1
A field $(\mathbb{K},+,\times)$ is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]] with respect to the multiplicative law $\times$.2
3
##### Remarks4
- Some authors (notably in the French literature) do not require a field to be commutative. The corresponding concept in English is called a [[division ring|division ring]].Revision 763
7/31/2026, 1:40:31 PM · Ancient Tree
Concept edited
Compare with revision 7423 changed lines
1
A field is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]].2
3
NB: In french, a field can be non-commutative. The corresponding concept in english is the one of [[division ring|division ring]].3
##### Remarks4
- Some authors (notably in the French literature) do not require a field to be commutative. The corresponding concept in English is called a [[division ring|division ring]].Revision 742
7/31/2026, 9:39:43 AM · Sequoia
Concept edited
Compare with revision 3812 changed lines
1
A field is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]].2
3
NB: In french, a field can be non-commutative. The corresponding concept in english is the one of [[division ring|division ring]].Revision 381
7/10/2026, 1:24:45 PM · Ancient Tree
Concept edited
Compare with revision 2552 changed lines
1
A field is a [[Ring|ring]] such that every nonzero element is invertible.1
A field is a [[Commutative ring|commutative ring]] such that every nonzero element is [[invertible|invertible]].Revision 255
7/7/2026, 9:27:47 AM · Ancient Tree
Concept edited
Compare with revision 1442 changed lines
1
A field is...1
A field is a [[Ring|ring]] such that every nonzero element is invertible.Revision 144
6/28/2026, 3:42:11 PM · Ancient Tree
Concept created
A field is...