Overview of Fields

Archimedean fields: ${\mathbf{Q}}, {\mathbf{R}}, {\mathbf{C}}$.
Nonarchimedean fields: everything else.
Global fields:
- Finite extension of ${\mathbf{Q}}$ or ${ \mathbf{F} }_p(t)$, or the function field of a geometrically integral curve over ${ \mathbf{F} }_{p^n}$
  - Equivalently: $\operatorname{ff}(A)$ for $A\in \mathsf{Alg} _{/ {{\mathbf{Z}}}} ^{{\mathrm{fg}}}$ with $A$ an integral domain and $\operatorname{krulldim}(A) = 1$.
Local fields:
- Finite extension of ${\mathbf{R}}, {\mathbf{Q}}, { {\mathbf{Q}}_p }, { \mathbf{F} }_p((t))$.
- ${\mathbf{R}}, {\mathbf{C}}, { \mathbf{F} }_{p^n}((t))$, or a finite extension of ${ {\mathbf{Q}}_p }$.
- ${\mathbf{R}}, {\mathbf{C}}, \operatorname{ff}(R)$ for $R$ a complete DVR with finite residue field
- $k$ a nondiscrete locally compact Hausdorff topological ring.
- $k$ the completion of a global field with respect to a nontrivial absolute value.

Archimedean and nonarchimedean

Idea: an ordered group $K$ is archimedean if it does not admit any infinitesimals or infinites: pairs $(x, y)$ such that $nx < y$ for all $n$, resp. $y$ such that $n1 < y$ for all $n$.

In the presence of an absolute value, $K$ is archimedean iff \begin{align*}x\in K \implies \exists n\text{ such that } {\left\lvert {nx} \right\rvert} > 1\end{align*} # Absolute values

An absolute value on a field $k$ is a map ${\left\lvert {{-}} \right\rvert}: k \rightarrow \mathbb{R}_{\geq 0}$ such that for all $x, y \in k$ the following hold:

$|x|=0$ if and only if $x=0$;
$|x y|=|x||y|$;
$|x+y| \leq|x|+|y|$.

If in addition the stronger ultrametric triangle inequality holds, \begin{align*}|x+y| \leq \max (|x|,|y|),\end{align*} then the absolute value is nonarchimedean; otherwise it is archimedean.

Two absolute values ${\left\lvert {{-}} \right\rvert}_1$ and ${\left\lvert {{-}} \right\rvert}_2$ on $K$ are equivalent if there exists an $\alpha \in \mathbb{R}_{>0}$ for which $|x|_2=|x|_1^{\alpha}$ for all $x \in k$.

Results

Every absolute value induces a metric $d(x, y) \coloneqq{\left\lvert {x-y} \right\rvert}$, but not every metric is induced this way.
Absolute values only exist for integral domains, and extend to fraction fields.
If $\operatorname{ch}K > 0$, every absolute value is nonarchimedean.
If ${ \mathbf{F} }_q$ is a finite field, there is only a trivial absolute value.
Get an absolute value from a valuation: \begin{align*}{\left\lvert {x} \right\rvert}_p \coloneqq p^{-v_p(x)},\qquad {\left\lvert {0} \right\rvert}_p \coloneqq p^{-\infty} = 0\end{align*}
If an integral domain is Cauchy complete with respect to an absolute value it is Cauchy complete with respect to all equivalent absolute values.

Topological Results

Weak approximation implies that two absolute values on the same field induce the same topology if and only if they are equivalent.
Translation is a homeomorphism. Thus in order to understand the topology of a topological group, we can focus on neighborhoods of the identity; a base of open neighborhoods about the identity determines the entire topology.

Examples

${\mathbf{C}},{\mathbf{R}},{\mathbf{Q}}$ with the Euclidean absolute value are archimedean.
Rational function fields $K { \left( {x} \right) } $ and formal Laurent series fields $K{\left(\left( x \right)\right) }$ are nonarchimedean.
${ {\mathbf{Z}}_{\widehat{p}} }$ and finite extensions of ${ {\mathbf{Q}}_p }$ are nonarchimedean.
The map ${\left\lvert {{-}} \right\rvert}: k \rightarrow \mathbb{R}_{\geq 0}$ defined by \begin{align*} |x|= \begin{cases}1 & \text { if } x \neq 0, \\ 0 & \text { if } x=0,\end{cases} \end{align*} is the trivial absolute value on $k$. It is nonarchimedean.
p-adic as an adic completion and a Cauchy completion
valuation ring of the adic completion of a rational Unsorted/function field over $K = { \mathbf{F} }_q$:

Links to this page

unresolved links output

geometrically integral in Unsorted/absolute value, -Unsorted/good reduction
ring of integers

absolute value

archimedean

For a nonarchimedean field with an absolute value ${\left\lvert {{-}} \right\rvert}$, say induced by a valuation, the ring of integers is given by the valuation ring, i.e. the closed disc: \begin{align*} {\mathcal{O}}_K \coloneqq\left\{{ x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} \leq 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) \geq 0}\right\} = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \subseteq K \end{align*} Its units are given by the boundary sphere: \begin{align*} {\mathcal{O}}_K^{\times}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} = 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) = 0}\right\} = {{\partial}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \end{align*} This is a local ring with maximal ideal the open disc: \begin{align*} {\mathfrak{m}}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} < 1 }\right\}= {\mathbb{D}}_K \end{align*}
number field

Has archimedean places and nonarchimedean places.
adic completion

Coincides with Cauchy completion when $R$ has a metric induced by a nonarchimedean absolute value.
Global field

The completion of a global field at a valuation / absolute value.

absolute value

The completion of $\operatorname{ff}(K)$ with respect to an absolute valueor valuation for $K$ a global field is a locally compact field, and thus a local field.
Cauchy completion

Tags: #todo #todo/stub Refs: absolute value
Archimedean place

Can then define an associated absolute value \begin{align*}{\left\lvert {x} \right\rvert}_v := \exp(v(x))\end{align*} where $e$ can be replaced with $c$ another constant.

The $p{\hbox{-}}$adic valuation $v_p: {\mathbf{Q}}\to {\mathbf{Z}}$ is defined using unique factorization in ${\mathbf{Q}}^{\times}$: \begin{align*} v_p(x) = v_p\left( \pm 1 \prod_{p_i\in \operatorname{Spec}{\mathbf{Z}}} p_i^{e_i} \right) \coloneqq e_i, \quad v_p(0) \coloneqq\infty \end{align*} with an associated nonarchimedean absolute value \begin{align*} {\left\lvert {x} \right\rvert}_p \coloneqq C^{-v_p(x)}, \quad c \in (0, 1),\quad {\left\lvert {0} \right\rvert}_p \coloneqq p^{-\infty}\coloneqq 0 \end{align*}