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## Definition: Field

A commutative ring $$R$$ is called a field, if $$R\neq 0$$ and if each element $$x\in R$$ with $$x\neq 0$$ has a multiplicative inverse.

“Unpacking” all definitions, this is equivalent with the following definition:

A set $$F$$ with two binary operations $$+$$ and $$\cdot$$, denoted by $$(F, + ,\cdot)$$, is called a field, if fulfills the following properties:

1. $$(F,+)$$ is an Abelian group.
2. If $$0$$ is the neutral element of $$(F,+)$$, then $$F^*:=(F \setminus \{0\},\cdot)$$ is an abelian group.
3. The distributivity law holds for all $$a,b,c\in F$$.

| | | | | created: 2014-03-28 21:11:15 | modified: 2016-09-04 21:29:59 | by: bookofproofs | references: [577], [6907]

## 2.Definition: Ordered Field

### CC BY-SA 3.0

[6907] Brenner, Prof. Dr. rer. nat., Holger: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014