Complete field
In
Constructions
Real and complex 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.
p-adic
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
Function field of a curve
For the function field of a curve every point corresponds to an
References
- OCLC 853269675.
See also
- 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 G 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 – concept in mathematics
- 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