We have seen that a ring is an abstract structure that maintains an abelian group structure with the addition operation but not necessarily with the multiplication operation. In order to build a structure that has all four basic mathematical operations (addition, subtraction, multiplication, division), we require a set of elements that contains an additive and a multiplicative abelian group. We call this structure a field.



A set of elements FF together with both operations

+:R×RR,(a,b)a+b, and .:R×RR,(a,b)ab\begin{aligned} +: R \times R \rightarrow R, &(a, b) \mapsto a+b, \quad \text { and } \\ .: R \times R \rightarrow R, &(a, b) \mapsto a \cdot b \end{aligned}

is called a field if the following properties are fulfilled:

  • R1: FF is an abelian group under addition (+).(+).

  • R2: F=F\{0}F^{*}=F \backslash\{0\} is an abelian group under multiplication ()(\cdot).

  • R3: The multiplication ()(\cdot) is distributive with respect to the addition (+)(+), i.e., for all a,b,cF:a, b, c \in F:

a(b+c)=ab+ac and (a+b)c=ac+bc.a \cdot(b+c)=a \cdot b+a \cdot c \text { and }(a+b) \cdot c=a \cdot c+b \cdot c .

Note: That the definition of a field implies that every nonzero element has a multiplicative inverse and hence is invertible.

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