Field

algebra

Definition

Körper

Field

A field $(F,+,\cdot )$ is a commutative ring with unity ($1\ne 0$) in which every non-zero element is invertible.

Formally, it satisfies the following properties:

1. $(F,+)$ is an Abelian group with neutral element $0$.
2. $(F\setminus \{0\},\cdot )$ is an Abelian group with neutral element $1$.
3. Multiplication distributes over addition.

$$\forall a\in F\setminus \{0\},\exists a^{-1}\in F:a\cdot a^{-1}=1$$