The $p$-adic Numbers

Introduction

The field of $p$-adic numbers is the simplest non-trivial example of everything in this Part: a field carrying a distance for which the triangle inequality is strengthened to the ultrametric inequality, a ring of integers which is compact and totally disconnected, and a completion which is not the Archimedean one. It is the first place where the general theory of Absolute Values, Valuations and Completions becomes concrete, and it is the base case of Local Fields: the field $\mathbb{Q}_p$ is the unique local field of characteristic zero with residue field $\mathbb{F}_p$, and every non-Archimedean local field of characteristic zero is a finite extension of some $\mathbb{Q}_p$.

This article develops the construction and the topology of $\mathbb{Z}_p$ and $\mathbb{Q}_p$ from the beginning: the inverse limit definition of $\mathbb{Z}_p$, its identification with the completion of $\mathbb{Z}$ in the $p$-adic absolute value, the extension of the absolute value to $\mathbb{Q}_p$, the digit expansion, Hensel's lemma, and the structure of the unit group. It states Ostrowski's theorem, which places the $p$-adic absolute values together with the Archimedean one as the only completions of $\mathbb{Q}$, and it develops the topological properties that the rest of the Part uses: compactness, total disconnectedness and local compactness of $\mathbb{Z}_p$ and $\mathbb{Q}_p$, the fundamental system of open subgroups, and the classification of the open and closed subsets. The article closes with the applications — the local field $\mathbb{Q}_p$ and its finite extensions, square roots and roots of unity, the square-classification of $\mathbb{Q}_p^\times$, and the role of $\mathbb{Q}_p$ as a completion of $\mathbb{Q}$ at a place — and with a precise statement of what is deferred.

The field $\mathbb{Q}_p$ is the topological object of this article; its analysis — the theory of continuous and analytic functions on $\mathbb{Q}_p$, the integration with respect to the Haar measure of Locally Compact Groups and Haar Measure in Part III, the $p$-adic logarithm and exponential as power series, the differential equations of $p$-adic analysis — belongs to that Part, where the measure and the limit are available. The construction of $\mathbb{Q}_p$ is that of Absolute Values, Valuations and Completions; the general theory of its finite extensions is that of Local Fields. Throughout, $p$ is a prime, $\lvert \cdot \rvert_p$ is the $p$-adic absolute value on $\mathbb{Q}$ normalised by $\lvert p \rvert_p = p^{-1}$, $v_p$ is the corresponding valuation, $\mathbb{Z}_p$ is the ring of $p$-adic integers and $\mathbb{Q}_p$ the field of $p$-adic numbers, with $\mathrm{M}_p = p\mathbb{Z}_p$ and residue field $\mathbb{F}_p$.

Construction

The Inverse Limit

Definition. The ring of $p$-adic integers is the inverse limit

$$ \mathbb{Z}_p = \varprojlim_{n} \mathbb{Z}/p^n\mathbb{Z} , $$

the transition maps $\mathbb{Z}/p^{n+1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$ being the reductions modulo $p^n$. It is a topological ring with the inverse limit topology, in which the sets $p^n\mathbb{Z}_p \subseteq \mathbb{Z}_p$ form a fundamental system of neighbourhoods of $0$.

Theorem. $\mathbb{Z}_p$ is a compact Hausdorff commutative topological ring, it is an integral domain, and it is a principal ideal domain with exactly one nonzero prime ideal, generated by $p$. Its field of fractions is $\mathbb{Q}_p$.

Proof. Each $\mathbb{Z}/p^n\mathbb{Z}$ is finite, hence compact and Hausdorff, and an inverse limit of compact Hausdorff spaces with continuous transition maps is compact and Hausdorff. Addition and multiplication are defined componentwise and are continuous, so $\mathbb{Z}_p$ is a topological ring. For the domain property, suppose $x y = 0$ with $x, y \neq 0$; choose $m,n$ minimal with $x_m \neq 0$ in $\mathbb{Z}/p^m$ and $y_n \neq 0$ in $\mathbb{Z}/p^n$; then $v_p(x) = m$ and $v_p(y) = n$ in the sense below, and the product has valuation $m+n$ with nonzero reduction modulo $p^{m+n+1}$, a contradiction. Every element of $\mathbb{Z}_p$ is either a unit or of the form $p^k u$ with $u$ a unit and $k \geq 1$: an element is a unit if and only if its reduction modulo $p$ is nonzero, and otherwise it lies in $p\mathbb{Z}_p$ and the argument repeats on the quotient. Hence every nonzero ideal is generated by some $p^k$, and $\mathbb{Z}_p$ is a principal ideal domain with unique maximal ideal $p\mathbb{Z}_p$. The fraction field is the localisation at $p$, and its elements are the $p$-adic numbers.

Example (the reduction map). The canonical map $\mathbb{Z}_p \to \mathbb{Z}/p^n\mathbb{Z}$ is surjective and continuous for the discrete topology on the target, and its kernel is $p^n\mathbb{Z}_p$. The finite quotients $\mathbb{Z}_p/p^n\mathbb{Z}_p \cong \mathbb{Z}/p^n\mathbb{Z}$ form a basis of neighbourhoods of $0$ by open ideals, so $\mathbb{Z}_p$ is a profinite ring; in particular the open subsets are generated by the cosets of the $p^n\mathbb{Z}_p$.

The $p$-adic Absolute Value

Definition. For $x \in \mathbb{Q}^\times$, write $x = p^{k}\tfrac{a}{b}$ with $a, b$ integers prime to $p$ and $k \in \mathbb{Z}$, and put $\lvert x \rvert_p = p^{-k}$, $\lvert 0 \rvert_p = 0$. The $p$-adic absolute value $\lvert \cdot \rvert_p$ is an absolute value on $\mathbb{Q}$ satisfying the strong triangle inequality

$$ \lvert x + y \rvert_p \leq \max(\lvert x \rvert_p, \lvert y \rvert_p) , $$

with equality whenever $\lvert x \rvert_p \neq \lvert y \rvert_p$. The associated valuation is $v_p(x) = -\log_p \lvert x \rvert_p$, the exponent of $p$ in $x$.

Theorem. The completion of $\mathbb{Q}$ with respect to $\lvert \cdot \rvert_p$ is the field $\mathbb{Q}_p$; the completion of $\mathbb{Z}$ with respect to the induced metric is $\mathbb{Z}_p$, and $\mathbb{Z}_p$ is the unit ball $\{x \in \mathbb{Q}_p : \lvert x \rvert_p \leq 1\}$, while $p\mathbb{Z}_p$ is its unique maximal ideal. The absolute value extends uniquely to $\mathbb{Q}_p$, the extension is complete, and $\mathbb{Q}$ embeds as a dense subfield.

Proof. The completion of a valued field is a valued field with the same value group completed; the value group of $\lvert \cdot \rvert_p$ is $p^{\mathbb{Z}}$, which is discrete and already complete as an ordered group, so the value group does not change. The completion of $\mathbb{Z}$ is $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ by the standard description of the completion of a ring in an ideal-adic metric, and this is $\mathbb{Z}_p$ by definition; the identification with the unit ball follows because $\mathbb{Z}$ is the unit ball of $\mathbb{Q}$ and the unit ball condition is closed under limits. Uniqueness of the extension to the completion is the uniqueness part of the general extension theorem for valued fields. Density is the definition of the completion.

Remark (Ostrowski's theorem). Every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to an Archimedean one, and there is exactly one up to equivalence, or to a $p$-adic one for a unique prime $p$. The equivalences classes are the places of $\mathbb{Q}$: the infinite place and the places $p$. The product formula

$$ \lvert x \rvert_\infty \prod_p \lvert x \rvert_p = 1 \qquad (x \in \mathbb{Q}^\times) $$

holds, with $\lvert \cdot \rvert_\infty$ the usual absolute value. Ostrowski's theorem is the arithmetic content of the statement that $\mathbb{Q}$ has one completion for each place, and it is the general field-theoretic statement of which the $p$-adic construction is the case $p$ prime.

Digit Expansions

Definition. A $p$-adic expansion is a series $\sum_{n=-N}^{\infty} a_n p^n$ with digits $a_n \in \{0, 1, \dots, p-1\}$; it converges in the $p$-adic absolute value because the terms tend to $0$ and $\mathbb{Q}_p$ is complete. Elements of $\mathbb{Z}_p$ are exactly those with $a_n = 0$ for $n < 0$.

Theorem (uniqueness of the expansion). Every $x \in \mathbb{Q}_p$ has exactly one $p$-adic expansion, and the assignment $x \mapsto (a_n)_{n \geq -N}$ is a bijection onto the set of digit sequences with finitely many negative indices. In particular $\mathbb{Z}_p$ is homeomorphic to the Cantor set $\{0, 1, \dots, p-1\}^{\mathbb{N}}$, and both $\mathbb{Z}_p$ and $\mathbb{Q}_p$ are uncountable.

Proof. Given $x$, define $x_n$ to be the reduction of $x$ modulo $p^n$ in $\mathbb{Z}/p^n\mathbb{Z}$; the $x_n$ are compatible and determine residues $a_n$ by $x_{n} - x_{n-1} = a_{n-1}p^{n-1}$. The partial sums converge to $x$ because the difference $x - \sum_{n < N} a_n p^n$ is divisible by $p^N$ and hence has absolute value at most $p^{-N}$. Uniqueness follows because a nonzero series with digits in the allowed range cannot converge to $0$. The homeomorphism with the product of the discrete finite sets $\{0,\dots,p-1\}$ follows from the description of the inverse limit, and uncountability from the uncountability of the product.

Example (negative integers). The expansion of $-1$ is $\sum_{n \geq 0}(p-1)p^n$; indeed the identity

$$ \sum_{n=0}^{N-1}(p-1)p^n = p^N - 1 $$

shows that the partial sums differ from $-1$ by $p^N$, so the series converges to $-1$ in the $p$-adic topology. This is the archetype of the difference between the Archimedean and the non-Archimedean intuition: an infinite sum of positive numbers with increasing Archimedean size converges because its terms tend to zero $p$-adically. The convergence of a $p$-adic series is a statement about the complete metric space $\mathbb{Q}_p$ and belongs to this article; the analytic theory of power series, the $p$-adic logarithm and exponential, and the integrals of $p$-adic analysis belong to Part III.

Example ($\mathbb{Z}_p$ versus the interval). The map $\mathbb{Z}_p \to \{0, \dots, p-1\}^{\mathbb{N}}$ is a homeomorphism onto a Cantor set; consequently $\mathbb{Z}_p$ has no connected subsets other than points, it has a basis of clopen sets, and every compact open subset of $\mathbb{Q}_p$ is a finite disjoint union of cosets of the form $a + p^n\mathbb{Z}_p$. This is the sharpest contrast with the interval, whose subsets have the intermediate value property.

Topology and Completeness

Fundamental Properties

Theorem. The following hold.

(a) $\mathbb{Z}_p$ is compact, complete, totally disconnected and profinite; it is the unit ball of $\mathbb{Q}_p$ and a compact open subring.

(b) $\mathbb{Q}_p$ is a locally compact, complete, non-Archimedean valued field; it is totally disconnected, and it is not compact.

(c) The sets $p^n\mathbb{Z}_p$, $n \geq 0$, form a fundamental system of open neighbourhoods of $0$ consisting of subgroups; the sets $a + p^n\mathbb{Z}_p$ with $n \in \mathbb{Z}$ form a basis of open sets of $\mathbb{Q}_p$ consisting of clopen sets.

(d) A subset of $\mathbb{Z}_p$ is compact if and only if it is closed, and a subset of $\mathbb{Q}_p$ is compact if and only if it is closed and bounded.

Proof. (a) Compactness and completeness are from the previous sections; total disconnectedness follows because $p^n\mathbb{Z}_p$ is an open subgroup with finite index, so the cosets are clopen and separate points; profiniteness is the description as an inverse limit of finite groups. The unit ball description is the previous theorem. (b) Similarly $\mathbb{Q}_p$ is covered by the translates $a + \mathbb{Z}_p$, $a \in \mathbb{Q}_p$ with $\lvert a \rvert_p \leq 1$, which are compact, so it is locally compact; it is complete; it is not compact because it contains the discrete sequence $p^{-n}$ with no convergent subsequence, the family $\{B(0, p^{-n})\}$ being an open cover with no finite subcover. (c) is a restatement of the definition of the $p$-adic topology, the sets being cosets of open subgroups. (d) In a locally compact metric space, closed subsets of compact sets are compact and compact subsets are closed; conversely a closed and bounded subset of $\mathbb{Q}_p$ is contained in some $p^{-N}\mathbb{Z}_p$, which is compact, hence is compact.

Remark (the absence of the intermediate value property). Because $\mathbb{Q}_p$ is totally disconnected, a continuous function $\mathbb{Z}_p \to \mathbb{Q}$ may be locally constant and still be non-trivial, and there is no analogue of the mean value theorem. This is why the analysis of Part III develops its own machinery — the measure, the integral and the theory of rigid analytic functions — rather than importing the real-variable theory.

Hensel's Lemma

Theorem (Hensel). Let $f \in \mathbb{Z}_p[X]$ and suppose that $a \in \mathbb{Z}_p$ satisfies

$$ f(a) \equiv 0 \pmod p, \qquad f'(a) \not\equiv 0 \pmod p . $$

Then there is a unique $b \in \mathbb{Z}_p$ with $f(b) = 0$ and $b \equiv a \pmod p$. More generally, if $\lvert f(a) \rvert_p < \lvert f'(a) \rvert_p^{\,2}$ then there is a unique $b$ with $f(b) = 0$ and $\lvert b - a \rvert_p < \lvert f'(a) \rvert_p$.

Proof. Define $a_0 = a$ and the Newton iteration

$$ a_{n+1} = a_n - \frac{f(a_n)}{f'(a_n)} . $$

The derivative $f'(a_n)$ is a unit of $\mathbb{Z}_p$ because $f'(a_n) \equiv f'(a) \not\equiv 0 \pmod p$, so the iteration is defined in $\mathbb{Z}_p$. Taylor expansion of $f$ at $a_n$ gives

$$ f(a_{n+1}) = \sum_{k \geq 2} \frac{f^{(k)}(a_n)}{k!}(a_{n+1} - a_n)^k , $$

the coefficients $f^{(k)}(a_n)/k!$ lying in $\mathbb{Z}_p$ because the Taylor expansion $f(X+Y) = \sum_k f^{(k)}(X)Y^k/k!$ of an integer polynomial has coefficients in $\mathbb{Z}_p[X,Y]$, so that each divided derivative $\frac{1}{k!}\frac{d^k}{dX^k}$ of a polynomial in $\mathbb{Z}_p[X]$ again has coefficients in $\mathbb{Z}_p$; hence $\lvert f(a_{n+1}) \rvert_p \leq \lvert a_{n+1} - a_n \rvert_p^{\,2} = \lvert f(a_n) \rvert_p^{\,2}$, and the sequence $(a_n)$ is Cauchy because consecutive differences are bounded by $\lvert f(a_n)\rvert_p$, which tends to $0$. The limit $b$ exists by completeness, and $f(b) = 0$ by continuity. For uniqueness, write $f(X) - f(Y) = (X - Y)g(X,Y)$ where $g(X,Y) = \sum_{k \geq 1} a_k \frac{X^k - Y^k}{X - Y}$ is a polynomial with integer coefficients satisfying $g(x,y) \equiv f'(a) \pmod p$ whenever $x \equiv y \equiv a \pmod p$. If $b, b' \in \mathbb{Z}_p$ are roots with $b \equiv b' \equiv a \pmod p$, then $0 = f(b) - f(b') = (b - b')g(b,b')$, and $g(b,b')$ is a unit of $\mathbb{Z}_p$ because it is congruent to the unit $f'(a)$ modulo $p$. Since $\mathbb{Z}_p$ is an integral domain, $b = b'$.

Corollary (square roots and roots of unity). Let $u \in \mathbb{Z}_p^\times$.

(a) If $u \bmod p$ is a square in $\mathbb{F}_p^\times$, then $u$ is a square in $\mathbb{Z}_p^\times$; for odd $p$ the converse holds, and for $p = 2$ an odd unit $u$ is a square in $\mathbb{Z}_2^\times$ if and only if $u \equiv 1 \pmod 8$.

(b) $\mathbb{Z}_p^\times$ contains exactly the $(p-1)$-st roots of unity, which reduce bijectively modulo $p$ and are the Teichmüller representatives of $\mathbb{F}_p^\times$.

Proof. (a) Apply Hensel to $f(X) = X^2 - u$ with $a$ a lift of a square root of $u \bmod p$; the derivative $2a$ is a unit for odd $p$, and for $p = 2$ a direct check of the unit group shows that the squares in $\mathbb{Z}_2^\times$ are exactly the units congruent to $1$ modulo $8$. (b) Apply Hensel to $f(X) = X^{p-1} - 1$ with $a$ a lift of a nonzero residue: $f'(a) = (p-1)a^{p-2}$ is a unit mod $p$, so each of the $p-1$ residues lifts uniquely to a root.

Example (a root of $-1$ in $\mathbb{Z}_5$). For $p = 5$ and $f(X) = X^2 + 1$ we have $f(2) = 5 \equiv 0 \pmod 5$ and $f'(2) = 4 \not\equiv 0 \pmod 5$, so Hensel produces a unique $b \in \mathbb{Z}_5$ with $b^2 = -1$ and $b \equiv 2 \pmod 5$. The Newton iteration $a_{n+1} = a_n - (a_n^2+1)/(2a_n)$ beginning at $a_0 = 2$ gives $a_1 = 3/4$, which is congruent to $7$ modulo $25$ because $3/4 - 7 = -25/4 \in 25\mathbb{Z}_5$; indeed $7^2 + 1 = 50 = 2 \cdot 5^2$, so $a_1$ is already a root modulo $25$; the successive iterates satisfy $\lvert f(a_n)\rvert_5 \leq 5^{-2^n}$, so the sequence converges to the root at a rate doubling the number of correct $p$-adic digits at each step. The computation is the standard illustration of the quadratic convergence of the Hensel iteration, and it can be checked modulo $5^{20}$ directly.

Arithmetic Consequences

The Unit Group

Theorem. The group $\mathbb{Q}_p^\times$ decomposes as a direct product

$$ \mathbb{Q}_p^\times \cong p^{\mathbb{Z}} \times \mathbb{Z}_p^\times , $$

and for $p$ odd the unit group decomposes as

$$ \mathbb{Z}_p^\times \cong \mu_{p-1} \times (1 + p\mathbb{Z}_p) , $$

with $\mu_{p-1}$ the group of $(p-1)$-st roots of unity and $1 + p\mathbb{Z}_p$ a pro-$p$ group; for $p = 2$,

$$ \mathbb{Z}_2^\times \cong \{\pm 1\} \times (1 + 4\mathbb{Z}_2) . $$

The maps are isomorphisms of topological groups when the factors carry the subspace topologies.

Proof. Every $x \in \mathbb{Q}_p^\times$ is uniquely $p^k u$ with $k = v_p(x)$ and $u \in \mathbb{Z}_p^\times$, giving the first decomposition; the topology is the product topology because $v_p$ is locally constant and the decomposition is continuous with continuous inverse. For the unit group, reduce modulo $p$: the residue map $\mathbb{Z}_p^\times \to \mathbb{F}_p^\times$ splits by Hensel's lemma, giving the Teichmüller subgroup $\mu_{p-1}$; the kernel is $1 + p\mathbb{Z}_p$, which is open and a pro-$p$ group because it is an inverse limit of the finite $p$-groups $(1+p\mathbb{Z}_p)/(1+p^n\mathbb{Z}_p)$ of $p$-power order. The intersection is trivial and the product is the whole group. For $p = 2$ the residue field has only one nonzero class, and the correct splitting uses $\{\pm 1\}$ together with $1 + 4\mathbb{Z}_2$: every odd $u$ satisfies $u \equiv 1 \pmod 4$ or $u \equiv 3 \pmod 4$, and in the second case $-u \equiv 1 \pmod 4$, so $u = \pm(1 + 4z)$ with $z \in \mathbb{Z}_2$, the two factors meeting only in $1$.

Remark. The decomposition is the topological input to the theory of characters of $\mathbb{Q}_p^\times$, which is part of the harmonic analysis of Part III: the dual of a product involves the duals of the factors, and the $p$-adic Fourier transform is a construction in the measure-theoretic setting. This article states only the group-theoretic and topological decomposition.

Finite Extensions and Local Fields

Definition. A non-Archimedean local field of characteristic zero is a field which is complete with respect to a discrete absolute value, with finite residue field, and of characteristic zero. By Local Fields every such field is a finite extension of some $\mathbb{Q}_p$, and conversely.

Theorem. Let $K/\mathbb{Q}_p$ be a finite extension of degree $n$. Then the absolute value of $\mathbb{Q}_p$ extends uniquely to $K$, the extension is complete, its valuation ring $\mathcal{O}_K$ is a free $\mathbb{Z}_p$-module of rank $n$ and is compact, and the residue field of $K$ is a finite extension of $\mathbb{F}_p$; the ramification index $e$ and the residue degree $f$ satisfy $ef = n$. The topology of $K$ is the product topology of $K \cong \mathbb{Q}_p^n$, and $K$ is locally compact and totally disconnected.

Proof. Uniqueness and completeness of the extension are the standard theorems for complete valued fields; the description of the valuation ring as a free $\mathbb{Z}_p$-module of rank $n$ is the structure theory of the integral closure of a valuation ring in a finite extension, and $ef = n$ is the standard fundamental identity of ramification theory. Local compactness follows because $\mathcal{O}_K$ is a compact open subring, and total disconnectedness is inherited from the residue field and the discrete valuation. These statements belong to Local Fields; they are recalled here to place $\mathbb{Q}_p$ among the local fields.

Example (the field $\mathbb{Q}_p$ inside its extensions). The unramified extension of degree $f$ over $\mathbb{Q}_p$ is obtained by adjoining a primitive $(p^f - 1)$-st root of unity, which exists in $\overline{\mathbb{Q}}_p$ and is integral, by Hensel's lemma applied to $X^{p^f-1} - 1$; this is the topological route to the finite fields and to the Frobenius. The totally ramified extensions are obtained by adjoining $p$-th roots of uniformisers, and the failure of some such roots to exist is exactly the condition that distinguishes $\mathbb{Q}_p$ from a perfectoid field, as in Perfectoid Spaces.

Summary

The ring of $p$-adic integers is $\mathbb{Z}_p = \varprojlim_n \mathbb{Z}/p^n\mathbb{Z}$, a compact, Hausdorff, profinite, totally disconnected commutative topological ring which is a principal ideal domain with unique maximal ideal $p\mathbb{Z}_p$; its field of fractions is the field of $p$-adic numbers $\mathbb{Q}_p$, which is the completion of $\mathbb{Q}$ for the $p$-adic absolute value $\lvert p \rvert_p = p^{-1}$. The absolute value satisfies the strong triangle inequality and extends uniquely to $\mathbb{Q}_p$; by Ostrowski's theorem the only nontrivial absolute values on $\mathbb{Q}$ are the Archimedean one and the $p$-adic ones, and the product formula $\lvert x \rvert_\infty \prod_p \lvert x \rvert_p = 1$ holds.

Every $x \in \mathbb{Q}_p$ has a unique $p$-adic expansion $\sum_{n \geq -N} a_n p^n$ with digits $0 \leq a_n < p$, the partial sums converging because the terms tend to zero; $\mathbb{Z}_p$ is homeomorphic to a Cantor set, is uncountable, and has a basis of clopen sets given by the cosets $a + p^n\mathbb{Z}_p$. $\mathbb{Q}_p$ is locally compact and complete but not compact, and a subset of $\mathbb{Q}_p$ is compact exactly when it is closed and bounded. Hensel's lemma gives the lift of a simple root modulo $p$ to a root in $\mathbb{Z}_p$, with quadratic convergence of the Newton iteration; its consequences include the liftability of square roots, the existence and uniqueness of the Teichmüller representatives $\mu_{p-1}$, and the decomposition $\mathbb{Q}_p^\times \cong p^{\mathbb{Z}} \times \mathbb{Z}_p^\times$ with $\mathbb{Z}_p^\times \cong \mu_{p-1} \times (1+p\mathbb{Z}_p)$ for odd $p$ and $\mathbb{Z}_2^\times \cong \{\pm 1\} \times (1+4\mathbb{Z}_2)$. Every non-Archimedean local field of characteristic zero is a finite extension of some $\mathbb{Q}_p$, with compact valuation ring a free $\mathbb{Z}_p$-module of rank the degree.

The construction, the topology and the algebraic structure of $\mathbb{Q}_p$ belong to this article; the $p$-adic analysis on $\mathbb{Q}_p$ — continuous and analytic functions, the integration with respect to the Haar measure of Locally Compact Groups and Haar Measure in Part III, the $p$-adic logarithm and exponential, and the differential equations over $\mathbb{Q}_p$ — belongs to that Part, where the measure and the limit are available.

Summary of Notation

Symbol	Meaning
$p$	A prime
$\lvert \cdot \rvert_p$	$p$-adic absolute value, $\lvert p \rvert_p = p^{-1}$
$v_p$	$p$-adic valuation, $v_p(x) = -\log_p \lvert x \rvert_p$
$\mathbb{Z}_p = \varprojlim_n \mathbb{Z}/p^n\mathbb{Z}$	Ring of $p$-adic integers
$\mathbb{Q}_p$	Field of $p$-adic numbers, the fraction field of $\mathbb{Z}_p$
$p\mathbb{Z}_p$	Unique maximal ideal of $\mathbb{Z}_p$, $\mathrm{M}_p$
$\mathbb{F}_p$	Residue field of $\mathbb{Z}_p$
$p^n\mathbb{Z}_p$	Fundamental system of open subgroups of $\mathbb{Z}_p$
$a + p^n\mathbb{Z}_p$	Basis of clopen sets of $\mathbb{Q}_p$
$\sum_{n \geq -N} a_n p^n$	$p$-adic expansion
$\mathbb{Z}_p^\times$, $\mathbb{Q}_p^\times$	Unit group and multiplicative group
$\mu_{p-1}$	Teichmüller representatives, the $(p-1)$-st roots of unity in $\mathbb{Z}_p$
$1 + p\mathbb{Z}_p$	Principal units, a pro-$p$ group
$\lvert \cdot \rvert_\infty$	The Archimedean absolute value on $\mathbb{Q}$
$\mathcal{O}_K$, $e$, $f$	Valuation ring of a finite extension and its ramification index and residue degree