Locally compact field
In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.[1] These kinds of fields were originally introduced in p-adic analysis since the fields are locally compact topological spaces constructed from the norm on . The topology (and metric space structure) is essential because it allows one to construct analogues of algebraic number fields in the p-adic context.
Structure
Finite dimensional vector spaces
One of the useful structure theorems for vector spaces over locally compact fields is that the finite dimensional vector spaces have only an equivalence class of norm: the sup norm[2] pg. 58-59.
Finite field extensions
Given a finite field extension over a locally compact field , there is at most one unique field norm on extending the field norm ; that is,
for all which is in the image of . Note this follows from the previous theorem and the following trick: if are two equivalent norms, and
then for a fixed constant there exists an such that
for all since the sequence generated from the powers of converge to .
Finite Galois extensions
If the index of the extension is of degree and is a Galois extension, (so all solutions to the minimal polynomial of any is also contained in ) then the unique field norm can be constructed using the field norm[2] pg. 61. This is defined as
Note the n-th root is required in order to have a well-defined field norm extending the one over since given any in the image of its norm is
since it acts as scalar multiplication on the -vector space .
Examples
Finite fields
All finite fields are locally compact since they can be equipped with the discrete topology. In particular, any field with the discrete topology is locally compact since every point is the neighborhood of itself, and also the closure of the neighborhood, hence is compact.
Local fields
The main examples of locally compact fields are the p-adic rationals and finite extensions . Each of these are examples of local fields. Note the algebraic closure and its completion are not locally compact fields[2] pg. 72 with their standard topology.
Field extensions of Qp
Field extensions can be found by using Hensel's lemma. For example, has no solutions in since
only equals zero mod if , but has no solutions mod . Hence is a quadratic field extension.
See also
- Compact group – Topological group with compact topology
- Complete field – algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback
- Local field – Locally compact topological field
- Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
- Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
- Ordered topological vector space
- Ramification of local fields
- Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
- Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
- Topological group – Group that is a topological space with continuous group action
- Topological module
- Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
- Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
- Topological vector space – Vector space with a notion of nearness
References
External links
- Inequality trick https://math.stackexchange.com/a/2252625