Completion of Fields
The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.
Incompleteness of the Rational Numbers
The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.
Consider the sequence of decimal approximations to $\sqrt2$:
$$ 1,\quad 1.4,\quad 1.41,\quad 1.414,\quad\ldots $$
The terms become arbitrarily close to one another, but the sequence does not converge inside $\mathbb{Q}$, because
$$ \sqrt2\notin\mathbb{Q}. $$
Thus the rational numbers contain “holes.” The process of filling these holes leads to the notion of completion.
Completion transforms a metric field into a complete metric field in which every Cauchy sequence converges.
This construction is fundamental both in analysis and number theory.
Metrics and Absolute Values
Let $K$ be a field equipped with an absolute value
$$ |\cdot|. $$
The associated metric is
$$ d(x,y)=|x-y|. $$
This metric defines convergence and Cauchy sequences.
A sequence
$$ (x_n) $$
is Cauchy if for every $\varepsilon>0$, there exists $N$ such that
$$ |x_n-x_m|<\varepsilon $$
whenever $m,n\ge N$.
Intuitively, the terms become arbitrarily close to one another.
A field is complete if every Cauchy sequence converges to an element of the field.
The rational numbers are incomplete with respect to both the ordinary absolute value and every $p$-adic absolute value.
Completion via Cauchy Sequences
The completion of a field is constructed from equivalence classes of Cauchy sequences.
Two Cauchy sequences
$$ (x_n), \qquad (y_n) $$
are considered equivalent if
$$ |x_n-y_n|\to0. $$
An element of the completion is such an equivalence class.
Addition and multiplication are defined termwise:
$$ (x_n)+(y_n)=(x_n+y_n), $$
$$ (x_n)(y_n)=(x_ny_n). $$
These operations are compatible with the equivalence relation.
The original field embeds naturally into its completion by identifying an element $a\in K$ with the constant sequence
$$ a,a,a,\ldots $$
The completion therefore enlarges the field while preserving its arithmetic structure.
Completion of $\mathbb{Q}$
The most familiar example is the completion of the rational numbers under the ordinary absolute value.
The resulting field is
$$ \mathbb{R}. $$
Every real number can be represented by a convergent Cauchy sequence of rationals.
For example, the irrational number $\sqrt2$ corresponds to the equivalence class of sequences approximating it.
The field $\mathbb{R}$ is complete, meaning every Cauchy sequence converges.
Completeness is one of the central properties underlying real analysis.
$p$-Adic Completion
If instead one uses the $p$-adic absolute value
$$ |\cdot|_p, $$
the completion of $\mathbb{Q}$ becomes the field
$$ \mathbb{Q}_p. $$
Thus the same rational numbers produce entirely different completed fields depending on the chosen notion of distance.
In the ordinary metric,
$$ p^n\to\infty. $$
In the $p$-adic metric,
$$ p^n\to0. $$
Consequently, sequences that diverge over the reals may converge $p$-adically.
The completion process therefore depends fundamentally on the underlying absolute value.
Dense Subfields
A subfield $K\subseteq L$ is dense if every element of $L$ can be approximated arbitrarily closely by elements of $K$.
The rational numbers are dense in both
$$ \mathbb{R} \qquad\text{and}\qquad \mathbb{Q}_p. $$
Thus every real or $p$-adic number can be approximated by rationals.
Density allows one to study complicated fields using simpler arithmetic approximations.
This principle underlies much of local analysis.
Topological Structure
Completion converts algebraic objects into topological objects.
The completed field inherits:
- a metric topology;
- notions of continuity;
- compactness properties;
- analytic structure.
For example, the field $\mathbb{R}$ is connected, while $\mathbb{Q}_p$ is totally disconnected.
This distinction reflects the fundamental geometric difference between archimedean and non-archimedean analysis.
In $\mathbb{R}$, intervals form continuous regions. In $\mathbb{Q}_p$, neighborhoods fragment into nested ultrametric balls.
Thus completions can produce radically different geometries from the same starting field.
Extensions of Absolute Values
Suppose $L/K$ is a field extension and $K$ carries an absolute value.
A natural question is whether the absolute value extends to $L$.
For example, the ordinary absolute value on $\mathbb{Q}$ extends uniquely to $\mathbb{R}$ and then to $\mathbb{C}$.
Similarly, the $p$-adic absolute value on $\mathbb{Q}$ extends to finite extensions of $\mathbb{Q}_p$.
The completed extension fields remain locally compact and complete.
This extension theory becomes central in local algebraic number theory.
Local Fields
Completions lead naturally to local fields.
Examples include:
$$ \mathbb{R}, \qquad \mathbb{C}, \qquad \mathbb{Q}_p. $$
Finite extensions of $\mathbb{Q}_p$ are also local fields.
These fields are “local” because they isolate arithmetic behavior near a single prime or infinite place.
By contrast, number fields such as
$$ \mathbb{Q}(\sqrt2) $$
are global fields.
Modern number theory studies global arithmetic by analyzing all local completions simultaneously.
Compactness and Local Compactness
The real numbers are locally compact: every point has a compact neighborhood.
The field $\mathbb{Q}_p$ is also locally compact.
In fact, the ring of $p$-adic integers
$$ \mathbb{Z}_p $$
is compact.
Compactness properties play a major role in harmonic analysis, representation theory, and adelic methods.
Many modern arithmetic theories rely on integrating functions over local fields.
Completion and Algebraic Closure
The completion of a field need not be algebraically closed.
For example:
- $\mathbb{R}$ is complete but not algebraically closed;
- $\mathbb{Q}_p$ is complete but not algebraically closed.
The algebraic closure of $\mathbb{R}$ is
$$ \mathbb{C}, $$
while the algebraic closure of $\mathbb{Q}_p$ is vastly more complicated.
Completing the algebraic closure of $\mathbb{Q}_p$ yields the field
$$ \mathbb{C}_p, $$
a $p$-adic analogue of the complex numbers.
These fields form the setting for $p$-adic analytic geometry.
Product Formula and Global Structure
The rational numbers possess many inequivalent completions:
$$ \mathbb{R}, \qquad \mathbb{Q}_2, \qquad \mathbb{Q}_3, \qquad \mathbb{Q}_5, \ldots $$
Each completion captures arithmetic information associated with one absolute value.
Together they satisfy the product formula:
$$ |x|_\infty\prod_p |x|_p=1. $$
Thus local completions are not independent. They fit into a global arithmetic structure.
This idea leads naturally to adeles and ideles.
Completion in Modern Number Theory
Completion is one of the foundational constructions of modern arithmetic.
It appears throughout:
- local field theory;
- Diophantine equations;
- algebraic geometry;
- automorphic forms;
- harmonic analysis;
- Galois representations.
Many global problems are first reduced to local problems in completions. Solutions are then assembled using local-global principles.
In this way, completion transforms arithmetic into geometry and analysis, providing the local perspective necessary for modern number theory.