Field

Definition

A field $F$ is a set equipped with two binary operations, addition ($+$) and multiplication ($\times$), that satisfy the following axioms:

1. Additive Structure:

   • $(F,+)$ forms an Abelian Group.
   • The additive identity is denoted as $0$

2. Multiplicative Structure:

   • $(F\{0\},\times )$ forms an abelian group
   • The multiplicative identity is denoted as $1$
   • $1\ne 0$; the multiplicative identity ($1$) is not equal to the additive identity ($0$)

3. Distributive Law:

   • For all $a,b,c\in F:a\times (b+c)=(a\times b)+(a\times c)$

More precisely, a field is a commutative division ring. In other words, a field is a Commutative Ring with unity where every nonzero element has a multiplicative inverse. A noncommutative division ring is called a “strictly skew field”.

Example

• The rational numbers ($\mathbb{Q}$)
• The real numbers ($\mathbb{R}$)
• The complex numbers ($\mathbb{C}$)