Summary

Field theory

Definition

A field is a set of elements that is closed under two operations, addition and multiplication.

Definition

A field is an ordered tuple $(F,+,\cdot )$ where $F$ is a set and $+$ and $\cdot$ are binary operations on $F$ such that:

• $(F,+)$ is an Abelian group.
• $(F\setminus \{0\},\cdot )$ is an Abelian group. Here, $0$ is the additive identity.
• The distributive laws hold: $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$ for all $a,b,c\in F$.

Intuitive explanation

Field models the concept of numbers. It is a set of numbers that is closed under addition and multiplication. The field axioms ensure that the field behaves like the set of rational numbers, any two elements can be added, subtracted, multiplied, and divided (except by 0) and still remain in the field.

Examples

• The set of rational numbers $\mathbb{Q}$ is a field with addition and multiplication.
• The set of real numbers $\mathbb{R}$ is a field with addition and multiplication.
• The binary field ${\mathbb{F}}_2=\{0,1\}$ with addition and multiplication modulo 2.