Field - Revision history

Vipul: 3 revisions

2008-05-12T16:20:27Z

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Vipul at 20:44, 20 January 2008

2008-01-20T20:44:27Z

← Older revision		Revision as of 20:44, 20 January 2008
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	{{applyingoperatorgives\|subring-closure\|integral domain}}		{{applyingoperatorgives\|subring-closure\|integral domain}}

			==Module theory==

			Modules over fields are precisely the same as vector spaces. In particular any finitely generated module over a field is a [[free module]] and the number of generators is independent of the choice of generating set.

Vipul at 17:17, 11 January 2008

2008-01-11T17:17:00Z

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 ==Definition==
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 * For all nonzero <math>a</math> in <math>F</math>, there exists a <math>b</math> in <math>F</math> such that <math>a * b = 1</math>
-[[Category: Basic definitions in commutative rings]]
+==Metaproperties==

Vipul: Started the page

2007-01-09T03:32:20Z

Started the page

New page

==Definition==

===Symbol-free definition===

A '''field''' is a [[commutative unital ring]] with the additional property that its [[multiplicative group]] comprises all the nonzero elements, that is, with the property that all nonzero elements are invertible.

Alternatively, a field is a [[commutative unital ring]] with no proper nontrivial ideal.

===Definition with symbols===

A '''field''' is a set <math>F</math> endowed with constants <math>0</math> and <math>1</math> (not equal), a unary operation <math>-</math> and binary operations <math>+</math> and <math>*</math> such that:

* <math>a + (b + c) = (a + b) + c</math> for all <math>a, b, c</math> in <math>F</math>
* <math>a + 0 = a</math> for all <math>a</math> in <math>F</math>
* <math>a + b = b + a</math> for all <math>a, b</math> in <math>F</math>
* <math>a + (-a) = 0</math> for all <math>a</math> in <math>F</math>
* <math>a * (b * c) = (a * b) * c</math> for all <math>a, b, c</math> in <math>F</math>
* <math>a * 1 = a</math> for all <math>a</math> in <math>F</math>
* <math>a * b = b * a</math> for all <math>a, b</math> in <math>F</math>
* <math>a * (b + c) = (a * b) + (a * c)</math> for all <math>a, b, c</math> in <math>F</math>
* For all nonzero <math>a</math> in <math>F</math>, there exists a <math>b</math> in <math>F</math> such that <math>a * b = 1</math>

[[Category: Basic definitions in commutative rings]]