A local field is a topological field which is Hausdorff and locally compact as a topological space.

Examples of local fields include:

• Any field together with the discrete topology.

• The field $\mathbb{R}$ of real numbers.

• The field $\mathbb{C}$ of complex numbers.

• The field $\mathbb{Q}_{p}$ of $p$–adic rationals, or any finite extension thereof.

• The field $\mathbb{F}_{q}((t))$ of formal Laurent series in one variable $t$ with coefficients in the finite field $\mathbb{F}_{q}$ of $q$ elements.

In fact, this list is complete—every local field is isomorphic as a topological field to one of the above fields.