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
The topology (and metric space structure) is essential because it allows one to construct analogues of algebraic number fields in the p-adic context.
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.
Given a finite field extension
over a locally compact field
, there is at most one unique field norm
extending the field norm
Note this follows from the previous theorem and the following trick: if
are two equivalent norms, and
then for a fixed constant
since the sequence generated from the powers of
If the index of the extension is of degree
is a Galois extension, (so all solutions to the minimal polynomial of any
) then the unique field norm
can be constructed using the field norm[2] pg.
Note the n-th root is required in order to have a well-defined field norm extending the one over
since it acts as scalar multiplication on the
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.
The main examples of locally compact fields are the p-adic rationals
Each of these are examples of local fields.
Note the algebraic closure
are not locally compact fields[2] pg.
72 with their standard topology.
Field extensions
can be found by using Hensel's lemma.
only equals zero mod
is a quadratic field extension.