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In mathematics, a **complete field** is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the *p*-adic numbers).

The real numbers are the field with the standard euclidean metric . Since it is constructed from the completion of with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field (since its absolute Galois group is ). In this case, is also a complete field, but this is not the case in many cases.

The p-adic numbers are constructed from by using the p-adic absolute value

where Then using the factorization where does not divide its valuation is the integer . The completion of by is the complete field called the p-adic numbers. This is a case where the field^{[1]} is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted

For the function field of a curve every point corresponds to an absolute value, or place, . Given an element expressed by a fraction the place measures the order of vanishing of at minus the order of vanishing of at Then, the completion of at gives a new field. For example, if at the origin in the affine chart then the completion of at is isomorphic to the power-series ring

- Completion (algebra) – in algebra, any of several related functors on rings and modules that result in complete topological rings and modules
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- Hensel's lemma – Result in modular arithmetic
- Henselian ring – local ring in which Hensel’s lemma holds
- Compact group – Topological group with compact topology
- Locally compact field
- Locally compact quantum group – relatively new C*-algebraic approach toward quantum groups
- Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
- Ordered topological vector space
- Ostrowski's theorem – On all absolute values of rational numbers
- Topological abelian group – topological group whose group is abelian
- Topological field – Algebraic structure with addition, multiplication, and division
- Topological group – Group that is a topological space with continuous group action
- Topological module
- Topological ring – ring where ring operations are continuous
- Topological semigroup – semigroup with continuous operation
- Topological vector space – Vector space with a notion of nearness