Fields

A field is a set containing at least two distinct elements called $0$ and $1$, along with operations of addition and multiplication satisfying these properties.

Thus, $\mathbb{R}$ and $\mathbb{C}$ are fields (${\mathbb{R}}^n$ is an n-dimensional space built on the field $\mathbb{R}$), as is the set of rational numbers along with the usual operations of addition and multiplication. Another example of a field is the set $\{0,1\}$ with the usual operations of addition and multiplication except that $1+1$ is defined to equal $0$.

Many of the definitions, theorems, and proofs in linear algebra that work for the fields $\mathbb{R}$ and $\mathbb{C}$ also work without change for arbitrary fields.