Locally compact field

Summary

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 edit

Finite dimensional vector spaces edit

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 edit

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 edit

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 edit

Finite fields edit

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 edit

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 edit

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 edit

References edit

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

External links edit

  • Inequality trick https://math.stackexchange.com/a/2252625