Fields

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.