Locally compact field

In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.{{citation|title=Functional Analysis and Valuation Theory|first=Lawrence|last=Narici|publisher=CRC Press|year=1971|isbn=9780824714840|pages=21–22|url=https://books.google.com/books?id=dSGMFF3viGkC&pg=PA21}}. These kinds of fields were originally introduced in p-adic analysis since the fields $\mathbb{Q}_p$ of p-adic numbers are locally compact topological spaces constructed from the norm $|\cdot|_p$ on $\mathbb{Q}$ . 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 one equivalence class of norms: the sup norm.{{Cite book|last=Koblitz|first=Neil|title=p-adic Numbers, p-adic Analysis, and Zeta-Functions|pages=57–74}} ^{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{N}

such that

$\left(\frac{\|x\|_1}{\|x\|_2} \right)^N < \frac{1}{c_1}$

for all

N \geq N_0

since the sequence generated from the powers of

N

converge to

0

== Finite Galois extensions ==

If the extension is of degree $n = [K:F]$ and $K/F$ is a Galois extension, (so all solutions to the minimal polynomial, or conjugate elements, of any $a \in K$ are also contained in $K$ ) then the unique field norm $|\cdot|_K$ can be constructed using the field norm ^{pg. 61}. This is defined as

$|a|_K = |N_{K/F}(a)|^{1/n}$

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

$N_{K/F}(f) = \det m_f = f^n$

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 $\mathbb{Q}_p$ and finite extensions $K/\mathbb{Q}_p$ . Each of these are examples of local fields. Note the algebraic closure $\overline{\mathbb{Q}}_p$ and its completion $\mathbb{C}_p$ are not locally compact fields ^{pg. 72} with their standard topology.

== Field extensions of Q<sub>p</sub> ==

Field extensions $K/\mathbb{Q}_p$ can be found by using Hensel's lemma. For example, $f(x) = x^2 - 7 = x^2 - (2 + 1\cdot 5 )$ has no solutions in $\mathbb{Q}_5$ since

$\frac{d}{dx}(x^2 - 5) = 2x$

only equals zero mod

p

x \equiv 0 \text{ } (p)

, but

x^2 - 7

has no solutions mod

5

. Hence

\mathbb{Q}_5(\sqrt{7})/\mathbb{Q}_5

is a quadratic field extension.

References

External links

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

Category:Topology