Locally compact field

In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. 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 an equivalence class of norm: the sup norm. ^{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_p

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 $p$ if $x\equiv 0{\text{ }}(p)$, but $x^{2}-7$ has no solutions mod $5$. Hence $\mathbb {Q} _{5}({\sqrt {7}})/\mathbb {Q} _{5}$ is a quadratic field extension.

See also

• Complete field – algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback
• Locally compact group – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
• Ramification of local fields
• Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
• Topological group – Group that is a topological space with continuous group action
• Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback

External links

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