Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C.

The rational numbers Q form a subgroup of the real numbers R under addition.

Consider the group of real numbers R under addition, and the subgroup Z of integers.

Let F denote an arbitrary field (such as the real numbers R or the complex numbers C).

For example, the set of real numbers R is a smooth manifold.

Archimedean two-dimensional local fields, which are formal power series over the real numbers R or the complex numbers C.

The real numbers R, together with addition as operation and its usual topology, form a topological group.

The real numbers R with the usual operations and ordering form a Euclidean field.

For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0.

Consider the positive real numbers R, a Lie group under the usual multiplication.