Field

definitionfield-theoryanalysis

Definition

A field $K$ is a ring in which the addition and multiplication operations are commutative, there exists a multiplicative identity, $1_K$, and every element $a\in K$ has a multiplicative inverse $a^{-1}\in K$ such that $a\cdot a^{-1}=1_K$. Simply, a field is a commutative division ring.

Alternatively, we could say that a set $K$ is a field if it has well-defined operations of addition, subtraction, multiplication, and division that behave similarly to those four operations in the rational numbers or the real numbers.

Remark

All fields are domains.

Examples