Fields
Learn fields and their properties in this lesson.
Introduction
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.
Field
Definition:
A set of elements $F$ together with both operations
$\begin{aligned} +: R \times R \rightarrow R, &(a, b) \mapsto a+b, \quad \text { and } \\ .: R \times R \rightarrow R, &(a, b) \mapsto a \cdot b \end{aligned}$
is called a field if the following properties are fulfilled:

R1: $F$ is an abelian group under addition $(+).$

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

R3: The multiplication $(\cdot)$ is distributive with respect to the addition $(+)$, i.e., for all $a, b, c \in F:$
$a \cdot(b+c)=a \cdot b+a \cdot c \text { and }(a+b) \cdot c=a \cdot c+b \cdot c .$
Note: That the definition of a field implies that every nonzero element has a multiplicative inverse and hence is invertible.
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