Locally compact field

In algebra, a locally compact field is a topological field whose topology forms a locally compact space (in particular, it is a Hausdorff space). These kinds of fields were originally introduced in p-adic analysis since the fields formula_1 are locally compact topological spaces constructed from the norm formula_2 on formula_3. 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 pg. 58-59.
Finite field extensions.
Given a finite field extension formula_4 over a locally compact field formula_5, there is at most one unique field norm formula_6 on formula_7 extending the field norm formula_8; that is,formula_9for all formula_10 which is in the image of formula_11. Note this follows from the previous theorem and the following trick: if formula_12 are two equivalent norms, andformula_13then for a fixed constant formula_14 there exists an formula_15 such thatformula_16for all formula_17 since the sequence generated from the powers of formula_18 converge to formula_19.
Finite Galois extensions.
If the index of the extension is of degree formula_20 and formula_4 is a galois extension, (so all solutions to the minimal polynomial of any formula_22 is also contained in formula_7) then the unique field norm formula_6 can be constructed using the field norm pg. 61. This is defined asformula_25Note the n-th root is required in order to have a well-defined field norm extending the one over formula_5 since given any formula_27 in the image of formula_11 its norm isformula_29since it acts as scalar multiplication on the formula_5-vector space formula_7.
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 formula_1 and finite extensions formula_33. Each of these are examples of local fields. Note the algebraic closure formula_34 and its completion formula_35 are not locally compact fields pg. 72 with their standard topology.
Field extensions of Qp.
Field extensions formula_33 can be found by using Hensel's lemma. For example, formula_37 has no solutions in formula_38 since formula_39only equals zero mod formula_40 if formula_41, but formula_42 has no solutions mod formula_43. Hence formula_44 is a quadratic field extension. 