User:Nuerk/Absolute values on fields

Let $$F$$ be a field. An absolute value on $$F$$ is a function $$\left|.\right|\colon\,F \to [0,\infty)$$ into the non-negative real numbers satisfying:




 * style="width: 200px" | $$\left|x\right| = 0 \iff x = 0$$
 * (Positive-definiteness)
 * $$\left|xy\right| = \left|x\right|\left|y\right|$$
 * (Multiplicativeness)
 * $$\left|x+y\right| \leq \left|x\right| + \left|y\right|$$
 * (Subadditivity)
 * }
 * $$\left|x+y\right| \leq \left|x\right| + \left|y\right|$$
 * (Subadditivity)
 * }

This generalises the real absolute value for arbitrary fields.

Basic properties
From the definition follows that:


 * style="width: 200px" |$$\left|1\right|=\left|-1\right|=\left|\zeta\right|=1$$
 * $$\zeta \in F$$ root of unity, i.e. $$\zeta^n=1$$ for some $$n \in \N,$$
 * $$\left|x\right|=\left|-x\right|$$
 * $$\forall x \in F,$$
 * $$\left|x^{-1}\right|=\left|x\right|^{-1}$$
 * $$\forall x \in F,$$
 * $$\left|\left|x\right|-\left|y\right|\right| \leq \left|x-y\right|$$
 * $$\forall x,y \in F, $$
 * $$\left|n\right| = \left|\sum_{i=1}^n 1 \right| \leq n$$
 * $$\forall n \in \N. $$
 * }
 * $$\forall x,y \in F, $$
 * $$\left|n\right| = \left|\sum_{i=1}^n 1 \right| \leq n$$
 * $$\forall n \in \N. $$
 * }
 * }

Examples

 * The trivial absolute value
 * $$\left|x\right| := \begin{cases}1 & x\ne0 \\0 & x=0\end{cases}$$
 * can be defined on any field. It will be excluded in the following discussion.


 * Given a prime number $$p$$ each rational number $$0 \neq x \in \Q$$ can be uniquely written as $$x=p^k\frac{a}{b}$$ with $$k,a \in \Z$$, $$b \in \N$$ and $$\operatorname{gcd}(p,ab)=\operatorname{gcd}(a,b)=1$$. Via setting
 * $$\left|x\right|_{p} =\left|p^k\frac{a}{b}\right|_{p}:=p^{-k}$$
 * an absolute value is defined, called the $$p$$-adic absolute value.


 * Let $$A$$ be a discrete valuation ring with uniformizing element $$\pi$$, i.e. $$\mathfrak{m}=\pi A$$, where $$\mathfrak{m}$$ is the unique maximal ideal of $$A$$. Each $$x \in \operatorname{Quot}(A)^\times$$ can be uniquely represented as $$x=\varepsilon\pi^k$$ with $$\varepsilon \in A^\times$$ and $$k \in \Z$$. Now
 * $$\left|x\right|:= \begin{cases}e^{-k} & x \in \operatorname{Quot}(A)^\times \\0 & x=0\end{cases}$$
 * defines an absolute value on the quotient field $$\operatorname{Quot}(A)$$.

Discrete absolute value
An absolute value $$\left|.\right|$$ on a field $$F$$ is called discrete if the image $$\left|F^\times\right| \subset \R^\times$$ of the group of units is a discrete subgroup of $$(\R^\times,\cdot)$$.

Archimedean absolute value
An absolute value $$\left|.\right|$$ on a field $$F$$ is called Archimedean if the set $$\left|\N\right|$$ is bounded. Otherwise the absolute value is called non-Archimedean. An absolute value $$\left|.\right|$$ is non-Archimedean if and only if it satisfies the strong triangle inequality
 * $$\left|x+y\right| \leq \operatorname{max}(\left|x\right|,\left|y\right|)$$.

In this case, it follows that
 * $$\left|x\right| \neq \left|y\right| \Rightarrow \left|x+y\right| = \operatorname{max}(\left|x\right|,\left|y\right|)$$.

For non-Archimedean absolute values the valuation ring $$A$$ of the field $$F$$ is defined by
 * $$A := \{x \in F : \left|x\right| \leq 1\}$$.

Since
 * $$\left|x\right| \leq 1$$ or $$\left|x^{-1}\right| \leq 1 \quad \forall x \in F$$,

it follows that $$F = \operatorname{Quot}(A)$$. This ring is local, i.e. it contains exactly one maximal ideal, namely
 * $$\mathfrak{m} = \{x \in F : \left|x\right| < 1\}$$.

The valuation ring is a principal ideal domain and thus a discrete valuation ring if and only if the corresponding absolute value is discrete.

Valuations
Let $$F$$ be a field. There is a 1-1 correspondence between non-Archimedean absolute values and valuations on $$F$$:

Given a non-Archimedean absolute value $$\left|.\right|$$ the function
 * $$v \colon\, F^\times \to \R \cup \{\infty\}, \; x \mapsto \begin{cases}-\operatorname{ln}(\left|x\right|) & x \neq 0 \\ \infty & x=0\end{cases}$$

is a valuation, i.e. it satisfies


 * $$ v(x) = \infty \iff x = 0$$
 * $$ v(xy) = v(x)+v(y)$$
 * $$ v(x+y) \geq \operatorname{min}(v(x),v(y))$$.
 * }
 * $$ v(x+y) \geq \operatorname{min}(v(x),v(y))$$.
 * }

Vice versa, given a valuation $$v$$,
 * $$\left|x\right|:=\begin{cases}e^{-v(x)} & x \ne 0\\0 & x=0\end{cases}$$

defines a non-Archimedean absolute value.

Topology and equivalence of absolute values
If $$\left|.\right|$$ is an absolute value on a field $$F$$, then
 * $$d(x,y):=\left|x-y\right| \quad \forall x,y \in F$$

defines a metric and therefore a topology on $$F$$.

Two absolute values $$\left|.\right|_{1}$$ and $$\left|.\right|_{2}$$ on $$F$$ are called equivalent, denoted by $$\left|.\right|_{1} \sim \left|.\right|_{2}$$, if they induce the same topology.

It can be proven that two absolute values are equivalent if and only if there exists an $$s>0$$ such that
 * $$\left|x\right|_{2} = \left|x\right|_{1}^s$$

for all $$x \in F$$.

Approximation theorem
Let $$\left|.\right|_{1},...,\left|.\right|_{n}$$ be pairwise inequivalent absolute values on a field $$F$$, let $$\varepsilon>0$$ and let $$a_{1},...,a_{n}\in F$$ be arbitrary elements. Then there exists $$x \in F$$ such that $$\left|x-a_{i}\right|_{i} < \varepsilon$$ for all $$i=1,...,n$$.

Ostrowski's theorem
A theorem proven by Alexander Ostrowski states that every absolute value on $$\Q$$ is either equivalent to the real absolute value or to the $$p$$-adic absolute value for a prime number $$p$$.

Discrete absolute values
Let $$F$$ be a field with discrete absolute value $$\left|.\right|$$ and let $$A$$ be the valuation ring with maximal ideal $$\mathfrak{m}$$. Then every ideal of $$A$$ is of the form $$\mathfrak{m}^n$$, $$n \in \N$$. The descending chain of ideals
 * $$A \supseteq \mathfrak{m} \supseteq \mathfrak{m}^2 \supseteq \mathfrak{m}^3 \supseteq ...$$

is a neighbourhood base at $$0$$ in $$A$$. For if $$\pi \in A$$ is a uniformizing element, in particular $$\left|\pi\right| < 1$$, then
 * $$\mathfrak{m}^n = \{x \in F : \left|x\right| < \left|\pi\right|^{n-1}\}$$.

Analogously, the chain
 * $$A^\times \supseteq U^{(1)} \supseteq U^{(2)} \supseteq U^{(3)} \supseteq ...$$

of subgroups
 * $$U^{(n)}:=1+\mathfrak{m}^n=\{x\in F^\times : \left|1-x\right| < \left|\pi\right|^{n-1}\}$$

is a neighbourhood basis of $$1$$ in $$F^\times$$. $$U^{(n)}$$ is called the $$n$$-th higher unit group and $$U^{(1)} = 1 + \mathfrak{m}$$ the group of principal units.

For every $$n \geq 1$$ we have the following isomorphisms:


 * $$\mathfrak{m}^n/\mathfrak{m}^{n+1} \cong A/\mathfrak{m}$$,
 * $$U^{(n)}/U^{(n+1)}\cong A/\mathfrak{m}$$,
 * $$A^\times/U^{(n)} \cong (A/\mathfrak{m}^n)^\times$$.
 * }
 * $$A^\times/U^{(n)} \cong (A/\mathfrak{m}^n)^\times$$.
 * }
 * }

Complete fields
Let $$F$$ be a valued field. $$F$$ is called complete if it is complete with respect to the induced metric, i.e. if every Cauchy sequence in $$F$$ converges.

Properties of complete fields
Let $$F$$ be a discretely valued field. Let $$A$$ be the valuation ring, $$\mathfrak{m}$$ the maximal ideal and $$\lambda_n\colon\,A/\mathfrak{m}^{n+1} \to A/\mathfrak{m}^n$$ the canonical projection for $$n \geq 1$$. If we equip the rings $$A/\mathfrak{m}^n$$ with the discrete topology and $$\prod_{n=1}^\infty A/\mathfrak{m}^n$$ with the product topology, then the projective limit
 * $$\varprojlim_{n}A/\mathfrak{m}^n = \{(x_n)_{n=1}^\infty \in \prod_{n=1}^{\infty} A/\mathfrak{m}^n : \lambda_n(x_{n+1}) = x_n\}$$

is a closed subset of the cartesian product and therefore a topological ring in a canonical way. The canonical functions
 * $$A \to \varprojlim_{n}A/\mathfrak{m}^n$$

and
 * $$A^\times \to \varprojlim_{n}A^\times/U^{(n)}$$

are both isomorphisms and homeomorphisms.

Furthermore, Hensel's lemma holds in complete fields:

Hensel's lemma
Let $$F$$ be complete with respect to a non-Archimedean absolute value $$\left|.\right|$$, let $$A=\{x \in F : \left|x\right| \leq 1\}$$ be the valuation ring with maximal ideal $$\mathfrak{m}=\{x \in F : \left|x\right| < 1\}$$ and let $$f \in A[x]$$ be a polynomial with $$0 \not\equiv f \bmod \, \mathfrak{m}$$. If the reduction
 * $$\bar{f} \in A/\mathfrak{m}[x]$$

has a decomposition
 * $$\bar{f}=\tilde{g}\tilde{h}$$,

into coprime polynomials $$\tilde{g},\tilde{h} \in A/\mathfrak{m}[x]$$, then there exist $$g,h \in A[x]$$ satisfying


 * $$f=gh$$,
 * $$\bar{g}=\tilde{g}$$,
 * $$\bar{h}=\tilde{h}$$,
 * $$\operatorname{deg}(g)=\operatorname{deg}(\tilde{g})$$.
 * }
 * $$\bar{h}=\tilde{h}$$,
 * $$\operatorname{deg}(g)=\operatorname{deg}(\tilde{g})$$.
 * }
 * }

Extensions of absolute values
Let $$F$$ be complete with respect to a non-Archimedean absolute value $$\left|.\right|$$ and let $$E/F$$ be a separable algebraic field extension. Then there exists a unique absolute value $$\left|\left|.\right|\right|$$ on $$E$$ which restricts to $$\left|.\right|$$ on $$F$$, i.e. satisfies $$\left|\left|x\right|\right| = \left|x\right|$$ for all $$x \in F$$.

If $$E/F$$ is a finite extension, $$[E:F]=n<\infty$$, then the extension $$\left|\left|.\right|\right|$$ is given by
 * $$\left|\left|\alpha\right|\right|=\left|N_{E/F}(\alpha)\right|^{\frac{1}{n}} \quad \forall \alpha \in E$$,

where $$N_{E/F}$$ is the field norm. In this case $$E$$ is again complete.

Completion
Every field $$F$$ with absolute value $$\left|.\right|_F$$ can be densely embedded in a complete field $$\widehat{F}$$. Let $$C$$ be the ring of Cauchy sequences in $$F$$ and $$N$$ the ideal of null sequences in $$F$$. Then $$\widehat{F} := C/N$$ is a field and $$\left|(x_{j})_{j \in \N}+N\right|_\widehat{F}:=lim_{j \to \infty}\left|x_{j}\right|_F$$ defines an absolute value on $$\widehat{F}$$. The field $$\widehat{F}$$ is complete with respect to this absolute value and $$F$$ can be regarded as a subfield of $$\widehat{F}$$ by virtue of the isometric embedding $$x \mapsto (x)_{n \in \N}+N$$.

If $$F$$ is already complete, then this embedding is an isomorphism.

The above completion coincides with the metric completion and is thus unique up to isometry.

Properties of completion

 * If $$A \subseteq F$$ (resp. $$\widehat{A} \subseteq \widehat{F}$$) is the valuation ring with respect to $$\left|.\right|$$ (resp. $$\widehat{\left|.\right|}$$), and $$\mathfrak{m}$$ (resp. $$\widehat{\mathfrak{m}}$$) is the maximal ideal, then
 * $$\widehat{A}/\widehat{\mathfrak{m}}\cong A/\mathfrak{m}$$.


 * Furthermore, if $$\left|.\right|$$ is discrete, then
 * $$\widehat{A}/\widehat{\mathfrak{m}}^n\cong A/\mathfrak{m}^n$$
 * for all $$n \geq 1$$.


 * Let $$\left|.\right|$$ be discrete, let $$R \subseteq A$$ be a complete set of equivalence class representatives for $$A/\mathfrak{m}$$ such that $$0 \in R$$, and let $$\pi \in A$$ be a uniformizer, then each $$0 \neq x \in F$$ can be uniquely represented as a convergent series
 * $$x=\pi^m\sum_{i=0}^{\infty} a_{i}\pi^i$$,
 * where $$a_i \in R$$, $$a_0\ne 0$$ and $$m\in \Z$$.

Examples

 * According to a theorem by Ostrowski, if $$F$$ is complete with respect to an Archimedean absolute value, then there exist an isomorphism $$\varphi \colon \, F \to \R$$ or $$\C$$ and a $$s \in (0,1]$$, such that
 * $$\left|x\right|=\left|\varphi(x)\right|^s \quad \forall x \in F$$.


 * Let $$p$$ be a prime number. The completion of the rational numbers $$\Q$$ with respect to the $$p$$-adic absolute value is the uncountable field $$\Q_{p}$$ of $p$-adic numbers given by
 * $$\Q_{p}=\left\{\sum_{j=k}^\infty a_{j}p^j : k \in \Z, a_{j} \in \{0,1,...,p-1\}\right\}$$.
 * Note, that the sequence of partial sums $$\left(\sum_{j=k}^n a_{j}p^j\right)_{n \in \N}$$ is a Cauchy sequence with respect to the $$p$$-adic absolute value.


 * A local field is a field $$F$$ which is complete with respect to a discrete absolute value and which has a finite residue class field. Local fields are locally compact.
 * Finite extensions of $$\Q$$ and $$\mathbb F_{p}(x)$$ are called global fields.
 * It can be proven that the local fields are precisely the completions finite extensions of the fields $$\Q_p$$ and the fields $$\mathbb F_{p}((x))$$ of formal Laurent series over $$\mathbb F_{p}$$.