Theorem: Construction of Fields from Integral Domains

Let \((R, +, \cdot)\) be an integral domain. Then there exists a unique field \((F, \ast, \circ)\) with the following properties:

(1) There is a subset \(S\subset F\), which is a ring isomorphic to \(R\), i.e. where \((S, \ast, \circ)\simeq (R, +, \cdot)\).

(2) \((F, \ast, \circ)\) is subfield of any other field \((X, \ast, \circ)\) fulfilling the property (1), i.e. \((F, \ast, \circ)\) is the minimal field with the property (1).

Proofs: 1

Parts: 1
Proofs: 2

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013