Locally compact field

In algebra, a locally compact field is a topological field whose topology forms a locally compact space[1] (in particular, it is a Hausdorff space). 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

References

  1. Narici, Lawrence (1971), Functional Analysis and Valuation Theory, CRC Press, pp. 21–22, ISBN 9780824714840.
  2. Koblitz, Neil. p-adic Numbers, p-adic Analysis, and Zeta-Functions. pp. 57–74.
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