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^{[2] pg. 58-59}.

Finite field extensions

Given a finite field extension ${\displaystyle K/F}$ over a locally compact field ${\displaystyle F}$, there is at most one unique field norm ${\displaystyle |\cdot |_{K}}$ on ${\displaystyle K}$ extending the field norm ${\displaystyle |\cdot |_{F}}$; that is,

${\displaystyle |f|_{K}=|f|_{F}}$

for all ${\displaystyle f\in K}$ which is in the image of ${\displaystyle F\hookrightarrow K}$. Note this follows from the previous theorem and the following trick: if ${\displaystyle ||\cdot ||_{1},||\cdot ||_{2}}$ are two equivalent norms, and

${\displaystyle ||x||_{1}<||x||_{2}}$

then for a fixed constant ${\displaystyle c_{1}}$ there exists an ${\displaystyle N_{0}\in \mathbb {N} }$ such that

${\displaystyle \left({\frac {||x||_{1}}{||x||_{2}}}\right)^{N}<{\frac {1}{c_{1}}}}$

for all ${\displaystyle N\geq N_{0}}$ since the sequence generated from the powers of ${\displaystyle N}$ converge to ${\displaystyle 0}$.

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 ${\displaystyle \mathbb {Q} _{p}}$ and finite extensions ${\displaystyle K/\mathbb {Q} _{p}}$. Each of these are examples of local fields. Note the algebraic closure ${\displaystyle {\overline {\mathbb {Q} }}_{p}}$ and its completion ${\displaystyle \mathbb {C} _{p}}$ are not locally compact fields^{[2] pg. 72} with their standard topology.