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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]

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

2.Definition: Ordered Field

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

Bibliography (further reading)

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

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