Locally compact field
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In algebra, a locally compact field is a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
whose topology forms a locally compact
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
.. These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p are locally compact topological spaces constructed from the norm , \cdot, _p on \mathbb. The topology (and metric space structure) is essential because it allows one to construct analogues of
algebraic number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a fi ...
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 In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the ...
pg. 58-59.


Finite field extensions

Given a finite field extension K/F over a locally compact field F, there is at most one unique field norm , \cdot, _K on K extending the field norm , \cdot, _F; that is,
, f, _K = , f, _F
for all f\in K which is in the image of F \hookrightarrow K. Note this follows from the previous theorem and the following trick: if , , \cdot, , _1,, , \cdot, , _2 are two equivalent norms, and
, , x, , _1 < , , x, , _2
then for a fixed constant c_1 there exists an N_0 \in \mathbb such that
\left(\frac \right)^N < \frac
for all N \geq N_0 since the sequence generated from the powers of N converge to 0.


Finite Galois extensions

If the index of the extension is of degree n = :F/math> and K/F is a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
, (so all solutions to the minimal polynomial of any a \in K is also contained in K) then the unique field norm , \cdot, _K can be constructed using the
field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K ...
pg. 61. This is defined as
, a, _K = , N_(a), ^
Note the n-th root is required in order to have a well-defined field norm extending the one over F since given any f \in K in the image of F \hookrightarrow K its norm is
N_(f) = \det m_f = f^n
since it acts as scalar multiplication on the F-vector space K.


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 \mathbb_p and finite extensions K/\mathbb_p. Each of these are examples of
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
s. Note the algebraic closure \overline_p and its completion \mathbb_p are not locally compact fields pg. 72 with their standard topology.


Field extensions of Qp

Field extensions K/\mathbb_p can be found by using
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
. For example, f(x) = x^2 - 7 = x^2 - (2 + 1\cdot 5 ) has no solutions in \mathbb_5 since
\frac(x^2 - 5) = 2x
only equals zero mod p if x \equiv 0 \text (p), but x^2 - 7 has no solutions mod 5. Hence \mathbb_5(\sqrt)/\mathbb_5 is a quadratic field extension.


See also

* * * * * * * * * * * * * * *


References

Topology


External links

* Inequality trick https://math.stackexchange.com/a/2252625 {{algebra-stub