**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:

- \((F,+)\) is an Abelian group.
- If \(0\) is the neutral element of \((F,+)\), then \(F^*:=(F \setminus \{0\},\cdot)\) is an abelian group.
- 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]

## 1.**Proposition**: $0$ is unequal $1$

## 2.**Definition**: Ordered Field

### CC BY-SA 3.0

[6907] **Brenner, Prof. Dr. rer. nat., Holger**: “Various courses at the University of OsnabrÃ¼ck”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

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