21.02.03 · number-theory / quadratic-forms-local-fields

$p$ -adic numbers $Q_{p}$ and $Z_{p}$

shipped3 tiersLean: none

Anchor (Master): Hensel 1897 *Jahresbericht DMV* 6, 83-88 (originator: $p$-adic numbers as formal power series in $p$); Hensel 1908 *Theorie der algebraischen Zahlen* (Teubner, Leipzig); Ostrowski 1916 *Acta Math.* 41, 271-284 (classification of absolute values on $\mathbb{Q}$); Serre *A Course in Arithmetic* Ch. 2; Cassels *Local Fields* (LMS Student Texts 3, 1986) Ch. 1-4; Neukirch *Algebraic Number Theory* (Grundlehren 322, 1999) Ch. II §§1-5; Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 1 §§1-3; Schikhof *Ultrametric Calculus* (Cambridge Studies 4, 1984) Ch. 1; Bachman *Introduction to p-adic Numbers and Valuation Theory* (Academic Press 1964); Robert *A Course in p-adic Analysis* (GTM 198, 2000) Ch. 1-2

Intuition Beginner

The ordinary absolute value $∣ x ∣$ measures size by distance from zero on the number line. The $p$ -adic absolute value $∣ x ∣_{p}$ for a fixed prime $p$ measures size by divisibility by $p$ . A number is small in the $p$ -adic sense when it is divisible by a large power of $p$ , and large when it is not divisible by $p$ at all. So $1000$ is small at the prime $5$ because $1000 = 5^{3} \cdot 8$ , and $7$ is large at the prime $5$ because $5$ does not divide $7$ .

Why do we bother with this? Because once you measure size this way, you get a whole new arithmetic with a real number system attached. The $p$ -adic numbers $Q_{p}$ form a complete system in which Cauchy sequences converge, just like the real numbers, but with the new size function $∣ \cdot ∣_{p}$ replacing the everyday absolute value. Every congruence question modulo $p^{n}$ in ordinary arithmetic is a question about $Z_{p}$ , the ring of $p$ -adic integers sitting inside $Q_{p}$ .

The reason this matters: a single rational number has many sizes. It has the ordinary size on the real line, and one $p$ -adic size for each prime. Number theory becomes the study of all these sizes at once. Local-to-global theorems say that a property holds for a number if and only if it holds at every prime and at infinity. The $p$ -adic numbers are the rigorous setting where the "at prime $p$ " half of that statement lives.

Visual Beginner

A schematic showing the integer $7$ being approximated by successively better $5$ -adic approximations $7, 7 + 0 \cdot 5, 7 + 0 \cdot 25, \dots$ on one axis, and on a parallel axis the same integer being approximated in the ordinary metric by $7, 7.0, 7.00, \dots$ . The two metrics measure distance in incompatible ways: in the $5$ -adic metric the numbers $5, 25, 125, 625$ are getting smaller and smaller (their distance to zero shrinks); in the ordinary metric they are getting larger.

The picture captures the essential point: $Q_{p}$ is the same set-theoretic completion procedure as $R$ , applied to a different way of measuring distance. Powers of $p$ shrink to zero in the $p$ -adic world. Cauchy sequences in this new metric have new limits, and those limits form $Q_{p}$ .

Worked example Beginner

Compute the $7$ -adic expansion of $- 1$ and of $1/3$ , and check that the geometric series $1 + p + p^{2} + p^{3} + \dots$ converges to $1/ (1 - p)$ in $Z_{p}$ .

Step 1a. The $7$ -adic expansion of $- 1$ . The claim is that $- 1 = 6 + 6 \cdot 7 + 6 \cdot 49 + 6 \cdot 343 + \dots$ , an infinite power series in $7$ with every digit equal to $6$ .

Step 1b. Check the partial sums. The first partial sum is $6$ ; adding $- 1$ gives $- 7$ , which is $- 1 \cdot 7$ . The second partial sum is $6 + 42 = 48$ ; adding $- 1$ gives $- 49 = - 1 \cdot 7^{2}$ . The third partial sum is $48 + 294 = 342$ ; adding $- 1$ gives $- 343 = - 1 \cdot 7^{3}$ . The pattern continues: the partial sum after $N$ terms differs from $- 1$ by $- 7^{N}$ , which has $7$ -adic absolute value $7^{- N}$ . The partial sums converge to $- 1$ in the $7$ -adic metric.

Step 2a. The $7$ -adic expansion of $1/3$ . We want a power series $a_{0} + a_{1} \cdot 7 + a_{2} \cdot 49 + \dots$ with each $a_{n} \in {0, 1, 2, 3, 4, 5, 6}$ summing to $1/3$ . Solve digit by digit.

Step 2b. Modulo $7$ : $1/3 \equiv 1 \cdot 5 = 5 (mod 7)$ , since $3 \cdot 5 = 15 = 2 \cdot 7 + 1 \equiv 1 (mod 7)$ . So $a_{0} = 5$ . Now $1/3 - 5 = - 14/3$ , and we want $- 14/3 = 7 \cdot (a_{1} + a_{2} \cdot 7 + \dots)$ , that is $- 2/3 = a_{1} + a_{2} \cdot 7 + \dots$ in $Z_{7}$ .

Step 2c. Modulo $7$ : $- 2/3 \equiv (- 2) \cdot 5 = - 10 \equiv 4 (mod 7)$ . So $a_{1} = 4$ . Continue: $- 2/3 - 4 = - 14/3$ , divide by $7$ again gives $- 2/3$ , and the cycle repeats. The digits are $5, 4, 4, 4, 4, \dots$ forever after the first.

Step 3. Geometric series in $Z_{p}$ . Consider the series $1 + p + p^{2} + p^{3} + \dots$ . The partial sums are $S_{N} = 1 + p + \dots + p^{N - 1} = (1 - p^{N}) / (1 - p)$ . The remainder $1/ (1 - p) - S_{N} = p^{N} / (1 - p)$ has $p$ -adic absolute value $p^{- N}$ (since $1 - p$ is a $p$ -adic unit, having absolute value $1$ ), which tends to zero as $N$ grows. So $S_{N} \to 1/ (1 - p)$ in $Z_{p}$ .

Step 4. Sanity check: with $p = 7$ , the formula gives $1/ (1 - 7) = - 1/6$ . We separately computed $- 1$ to be $6 + 6 \cdot 7 + 6 \cdot 49 + \dots = 6 \cdot (1 + 7 + 49 + \dots)$ , so $1 + 7 + 49 + \dots = - 1/6$ . The two computations agree.

What this tells us: the $p$ -adic numbers are not just a formal device. They support genuine series convergence, genuine arithmetic with negative numbers (every negative integer has a "positive" $p$ -adic expansion), and genuine fractions (every rational with denominator coprime to $p$ sits inside $Z_{p}$ , and the residual ones differ only by a power of $p$ ). The $p$ -adic expansion of any element is unique.

Check your understanding Beginner

Formal definition Intermediate+

Fix a rational prime $p$ . The $p$ -adic valuation on $Z ∖ {0}$ is the function $v_{p} : Z ∖ {0} \to Z_{\geq 0}$ assigning to each non-zero integer $n$ the largest exponent $v_{p} (n)$ such that $p^{v_{p} (n)}$ divides $n$ . Extend to $Q^{\times}$ multiplicatively by $v_{p} (a / b) = v_{p} (a) - v_{p} (b)$ , and to all of $Q$ by $v_{p} (0) = + \infty$ . The valuation satisfies $v_{p} (x y) = v_{p} (x) + v_{p} (y)$ and $v_{p} (x + y) \geq min (v_{p} (x), v_{p} (y))$ for every $x, y \in Q$ .

The $p$ -adic absolute value is $∣ x ∣_{p} := p^{- v_{p} (x)}$ for $x \neq = 0$ and $∣0 ∣_{p} = 0$ . The four signature properties hold: $∣ x ∣_{p} \geq 0$ with equality only at $x = 0$ , $∣ x y ∣_{p} = ∣ x ∣_{p} ∣ y ∣_{p}$ , the strong triangle inequality $∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ , and consequently the ordinary triangle inequality.

The $p$ -adic metric on $Q$ is $d_{p} (x, y) := ∣ x - y ∣_{p}$ . It makes $Q$ into a metric space whose Cauchy sequences and topology differ from the Euclidean ones.

Definition (the two constructions of $Z_{p}$ ). Each of the following defines a complete topological ring, and the two are canonically isomorphic.

Construction A (inverse limit). Set $$ \mathbb{Z}p := \varprojlim_n \mathbb{Z}/p^n\mathbb{Z} = {(x_n){n \geq 1} : x_n \in \mathbb{Z}/p^n\mathbb{Z}, ; x_{n+1} \equiv x_n \pmod{p^n}}. $$ This is a ring under componentwise operations, topologised as a closed subset of the product $\prod_{n} Z / p^{n} Z$ with each factor discrete.

Construction B (metric completion). Set $$ \mathbb{Z}_p := \widehat{\mathbb{Z}}^{|\cdot|_p} = {[x_n] : x_n \in \mathbb{Z} \text{ Cauchy in } |\cdot|_p} / (\text{null Cauchy sequences}), $$ the completion of $Z$ at the metric $d_{p}$ , taken as the ring of equivalence classes of Cauchy sequences modulo those tending to zero.

The $p$ -adic numbers $Q_{p}$ are the fraction field of $Z_{p}$ , equivalently the metric completion of $Q$ at $∣ \cdot ∣_{p}$ . The valuation extends uniquely to $Q_{p}$ by continuity, and $Z_{p} = {x \in Q_{p} : ∣ x ∣_{p} \leq 1}$ .

Notation. The unit group is $Z_{p}^{\times} = {x \in Z_{p} : ∣ x ∣_{p} = 1}$ , the maximal ideal is $p Z_{p} = {x \in Z_{p} : ∣ x ∣_{p} < 1}$ , and the residue field is $Z_{p} / p Z_{p} ≅ F_{p}$ . Every $x \in Q_{p}^{\times}$ admits a unique decomposition $x = p^{v} u$ with $v = v_{p} (x) \in Z$ and $u \in Z_{p}^{\times}$ .

Counterexamples to common slips

The absolute value $∣ x ∣_{p}$ is not the count of factors of $p$ — it is $p^{- v_{p} (x)}$ . Doubling the count of factors halves $∣ x ∣_{p}$ multiplicatively in the base $p$ .
The strong triangle inequality forces every triangle in $Q_{p}$ to be isosceles: if $∣ x ∣_{p} \neq = ∣ y ∣_{p}$ , then $∣ x + y ∣_{p} = max (∣ x ∣_{p}, ∣ y ∣_{p})$ . This fails in the Euclidean metric, where $∣1 + 1∣ = 2 = ∣1∣ + ∣1∣$ rather than $max (∣1∣, ∣1∣) = 1$ .
An open ball $B (a, r) = {x : ∣ x - a ∣_{p} < r}$ in $Q_{p}$ is also closed, and every one of its points is a centre. Euclidean intuition that "open ball" and "closed ball" are different fails here.

Key theorem with proof Intermediate+

Theorem (Equivalence of the two constructions; Serre Ch. II §1 Proposition 2). The inverse-limit ring $lim_{n} Z / p^{n} Z$ and the metric-completion ring $Z^{∣ \cdot ∣_{p}}$ are canonically isomorphic as topological rings.

Proof. Write $A = lim_{n} Z / p^{n} Z$ for the inverse limit and $B = Z^{∣ \cdot ∣_{p}}$ for the metric completion. Construct mutually inverse continuous ring homomorphisms $Φ : A \to B$ and $Ψ : B \to A$ .

Map $Φ : A \to B$ . Given $(x_{n}) \in A$ with $x_{n} \in Z / p^{n} Z$ and $x_{n + 1} \equiv x_{n} (mod p^{n})$ , choose integer lifts $\tilde{x}_{n} \in Z$ of each $x_{n}$ subject to $0 \leq \tilde{x}_{n} < p^{n}$ (the canonical lift). The compatibility $x_{n + 1} \equiv x_{n} (mod p^{n})$ gives $\tilde{x}_{n + 1} - \tilde{x}_{n} \in p^{n} Z$ , so $∣ \tilde{x}_{n + 1} - \tilde{x}_{n} ∣_{p} \leq p^{- n}$ . For $m \leq n$ , telescoping yields $∣ \tilde{x}_{n} - \tilde{x}_{m} ∣_{p} \leq p^{- m}$ , so $(\tilde{x}_{n})$ is a Cauchy sequence in $Z$ at $∣ \cdot ∣_{p}$ . Set $Φ ((x_{n})) := [(\tilde{x}_{n})] \in B$ . The choice of canonical lifts makes $Φ$ well-defined; any other compatible choice differs by a null Cauchy sequence and gives the same class in $B$ .

Map $Ψ : B \to A$ . Given a Cauchy sequence $(y_{k})$ in $Z$ at $∣ \cdot ∣_{p}$ , for each $n \geq 1$ the sequence of residues $(y_{k} mod p^{n})$ is eventually constant: there is some $K_{n}$ such that $∣ y_{k} - y_{K_{n}} ∣_{p} \leq p^{- n}$ for all $k \geq K_{n}$ , that is $y_{k} \equiv y_{K_{n}} (mod p^{n})$ for all $k \geq K_{n}$ . Call this eventual residue $y_{\infty}^{(n)} \in Z / p^{n} Z$ . The compatibility $y_{\infty}^{(n + 1)} \equiv y_{\infty}^{(n)} (mod p^{n})$ holds because both come from sufficiently late terms in the same Cauchy sequence. Set $Ψ ([(y_{k})]) := (y_{\infty}^{(n)})_{n} \in A$ . Null Cauchy sequences map to $(0, 0, \dots)$ , so $Ψ$ is well-defined on equivalence classes.

Inverse property. Check $Ψ \circ Φ = id_{A}$ : given $(x_{n}) \in A$ with canonical lifts $\tilde{x}_{n}$ , the Cauchy sequence $(\tilde{x}_{k})$ has eventual residue modulo $p^{n}$ equal to $x_{n}$ itself (since $\tilde{x}_{k} \equiv x_{n} (mod p^{n})$ for all $k \geq n$ by the compatibility), so $Ψ (Φ ((x_{n}))) = (x_{n})$ . Check $Φ \circ Ψ = id_{B}$ : given $[(y_{k})] \in B$ , the inverse-limit element $(y_{\infty}^{(n)})$ has canonical lifts $\tilde{y}_{\infty}^{(n)} \in {0, \dots, p^{n} - 1}$ , and the Cauchy sequence $(\tilde{y}_{\infty}^{(n)})$ has the same equivalence class as $(y_{k})$ since both eventually agree modulo $p^{n}$ for each $n$ .

Topological compatibility. The inverse-limit topology on $A$ has fundamental open neighbourhoods of $0$ given by the kernels $p^{n} A = ker (A \to Z / p^{n} Z)$ . The metric topology on $B$ has fundamental open neighbourhoods of $0$ given by ${[(y_{k})] : ∣ y_{k} ∣_{p} < p^{- n} eventually}$ . Under $Φ$ , an element $(x_{n}) \in p^{n} A$ has $x_{1} = \dots = x_{n} = 0$ , hence canonical lifts $\tilde{x}_{1} = \dots = \tilde{x}_{n} = 0$ and $∣ \tilde{x}_{k} ∣_{p} \leq p^{- n}$ for all $k$ , putting $Φ ((x_{n}))$ inside the $p^{- n}$ -ball of $B$ . Conversely, the eventual-residue construction $Ψ$ sends the $p^{- n}$ -ball of $B$ inside $p^{n} A$ . So $Φ$ and $Ψ$ are continuous in both directions, identifying the two topologies.

Ring structure. Both $Φ$ and $Ψ$ preserve sums and products: componentwise operations on the inverse limit match termwise operations on Cauchy sequences once lifts are chosen. The verification reduces to the compatibility $\tilde{x}_{n} + \tilde{y}_{n} \equiv x + y_{n} (mod p^{n})$ , which is automatic. $□$

Bridge. The equivalence of constructions builds toward 21.02.04 Hensel's lemma and the broader theory of unramified extensions, where lifting roots from $F_{p}$ to $Z_{p}$ requires both pictures at once: the inverse-limit picture lets one work degree by degree, while the metric picture supplies the Newton-iteration convergence. The foundational reason both constructions agree is exactly that the $p$ -adic topology on $Z$ is determined by the kernel filtration $p^{n} Z$ , and this is the same filtration whether one quotients then takes the limit or completes the metric directly. This is exactly the local-to-global pattern that appears again in 21.03.01 Riemann zeta, where the Euler product over primes $\prod_{p} (1 - p^{- s})^{- 1}$ expresses the global zeta function as a product of local pieces, one per prime, and the local piece at $p$ is built from $Q_{p}$ -arithmetic. The central insight is that completion in a metric and inverse limit of finite quotients are two presentations of the same construction, generalising to any commutative ring with a filtration by ideals. Putting these together, the foundational reason is that the two constructions are isomorphic identifies $p$ -adic integers with formal power series in $p$ , the bridge between number-theoretic and analytic perspectives. The same equivalence is dual to the Tate-module construction in 21.05.01 $ℓ$ -adic Galois representations, where the Tate module of an elliptic curve $T_{ℓ} E = lim_{n} E [ℓ^{n}]$ uses the inverse-limit recipe at every prime $ℓ$ .

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the strong triangle inequality $∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ becomes an equality whenever $∣ x ∣_{p} \neq = ∣ y ∣_{p}$ .

Hint

Suppose without loss $∣ x ∣_{p} > ∣ y ∣_{p}$ . Use the strong triangle inequality applied to $x = (x + y) - y$ to derive a lower bound on $∣ x + y ∣_{p}$ .

Answer

Assume $∣ x ∣_{p} > ∣ y ∣_{p}$ . Set $z = x + y$ , so $x = z - y$ . The strong triangle inequality gives $∣ x ∣_{p} = ∣ z - y ∣_{p} \leq max (∣ z ∣_{p}, ∣ y ∣_{p})$ . If $∣ z ∣_{p} \leq ∣ y ∣_{p}$ , then this maximum is $∣ y ∣_{p}$ , but $∣ x ∣_{p} > ∣ y ∣_{p}$ , contradiction. So $∣ z ∣_{p} > ∣ y ∣_{p}$ , and $∣ x ∣_{p} \leq ∣ z ∣_{p}$ . Combined with the upper bound $∣ z ∣_{p} = ∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p}) = ∣ x ∣_{p}$ , we get $∣ x + y ∣_{p} = ∣ x ∣_{p} = max (∣ x ∣_{p}, ∣ y ∣_{p})$ . Rubric: full credit for the symmetric argument plus the strict-inequality case identification.

Exercise 4 (medium, symbolic).

Prove that $Z_{p}^{\times} = {x \in Z_{p} : ∣ x ∣_{p} = 1}$ and that the quotient $Z_{p} / p Z_{p}$ is isomorphic to $F_{p}$ as a ring.

Hint

For the first part, multiplication of absolute values shows that $∣ x y ∣_{p} = 1$ when $∣ x ∣_{p} = ∣ y ∣_{p} = 1$ . For the second, the inverse-limit description gives $Z_{p} \to Z / p Z$ projected to the first component.

Answer

If $x \in Z_{p}^{\times}$ with inverse $y$ , then $∣ x ∣_{p} ∣ y ∣_{p} = ∣ x y ∣_{p} = ∣1 ∣_{p} = 1$ with both factors $\leq 1$ , forcing both to equal $1$ . Conversely if $∣ x ∣_{p} = 1$ , then $x \in Z_{p}$ is non-zero, and a $p$ -adic expansion $x = a_{0} + a_{1} p + \dots$ with $a_{0} \in {1, \dots, p - 1}$ shows $x$ has a multiplicative inverse: working modulo successive powers $p^{n}$ , the residue of $x$ in $(Z / p^{n} Z)^{\times}$ is invertible (since $a_{0} \neq = 0$ in $F_{p}$ which is a field), and the compatible inverses define an inverse in the inverse limit. For the second part, the projection $Z_{p} \to Z / p Z ≅ F_{p}$ from the inverse-limit description is a ring homomorphism with kernel exactly $p Z_{p}$ (the set of elements whose first digit is zero), giving $Z_{p} / p Z_{p} ≅ F_{p}$ . Rubric: full credit for both directions of the unit characterisation and the explicit isomorphism.

Exercise 5 (medium, short-answer).

Show that $Z_{p}$ is compact in the $p$ -adic topology.

Hint

Use the inverse-limit description: $Z_{p}$ is a closed subset of the product $\prod_{n} Z / p^{n} Z$ of finite discrete spaces. Tychonoff or sequential compactness applies.

Answer

Realise $Z_{p} = lim_{n} Z / p^{n} Z$ as a subspace of $\prod_{n} Z / p^{n} Z$ with each factor finite and discrete. The product is compact by Tychonoff's theorem (a product of finite spaces is compact); the inverse-limit subspace is defined by the closed compatibility conditions ${(x_{n}) : x_{n + 1} mod p^{n} = x_{n}}$ for each $n$ . A closed subspace of a compact space is compact, so $Z_{p}$ is compact. Alternatively: $Z_{p}$ is sequentially compact, since every sequence in $Z_{p}$ has a convergent subsequence by a diagonal argument on the residues modulo $p^{n}$ . Rubric: full credit for either the Tychonoff or the sequential argument.

Exercise 7 (hard, short-answer).

Prove that $Z_{p}$ is totally disconnected: its only connected subsets are singletons.

Hint

Suppose $C \subseteq Z_{p}$ contains two distinct points $x \neq = y$ . Show that the strong triangle inequality lets you split $C$ into two non-empty disjoint open subsets via an open ball.

Answer

Suppose $C \subseteq Z_{p}$ is connected and contains $x \neq = y$ . Set $r = ∣ x - y ∣_{p} > 0$ . The open ball $B (x, r) = {z : ∣ z - x ∣_{p} < r}$ contains $x$ but not $y$ . In an ultrametric space, every open ball is also closed: if $z_{n} \in B (x, r)$ and $z_{n} \to z$ , then $∣ z - x ∣_{p} \leq max (∣ z - z_{n} ∣_{p}, ∣ z_{n} - x ∣_{p}) < r$ for $n$ large enough, so $z \in B (x, r)$ . Then $C = (C \cap B (x, r)) ⊔ (C ∖ B (x, r))$ is a separation of $C$ into two non-empty disjoint clopen subsets, contradicting connectedness. So $C$ contains at most one point, that is $C$ is a singleton. Rubric: full credit for the open-and-closed argument plus the separation.

Exercise 8 (hard, short-answer).

Show that the series $\sum_{n \geq 1} n! \cdot p^{n}$ converges in $Z_{p}$ , but the same series with $p^{n}$ replaced by $1$ does not converge in $Z_{p}$ .

Hint

A series $\sum a_{n}$ converges in $Z_{p}$ if and only if $∣ a_{n} ∣_{p} \to 0$ . Compute the $p$ -adic absolute value of $n! \cdot p^{n}$ versus $n!$ alone.

Answer

A key feature of the ultrametric absolute value is that $\sum a_{n}$ converges in $Z_{p}$ if and only if the partial sums are Cauchy, which by the strong triangle inequality is equivalent to $∣ a_{n} ∣_{p} \to 0$ (since $∣ S_{m} - S_{n} ∣_{p} = ∣ a_{n + 1} + \dots + a_{m} ∣_{p} \leq max_{n < k \leq m} ∣ a_{k} ∣_{p}$ ). Now $v_{p} (n! \cdot p^{n}) = v_{p} (n!) + n \geq n$ since $v_{p} (n!) \geq 0$ , so $∣ n! \cdot p^{n} ∣_{p} \leq p^{- n} \to 0$ and the series converges. By contrast $v_{p} (n!) = \sum_{k \geq 1} ⌊ n / p^{k} ⌋$ (Legendre's formula) grows like $n / (p - 1)$ , so $v_{p} (n!) \to \infty$ and $∣ n! ∣_{p} \to 0$ as well — actually this means $\sum n!$ does converge in $Z_{p}$ . Revisiting: the precise comparison shows both series converge, but at different rates. To produce a non-convergent example, use $\sum 1$ (constant term): $∣1 ∣_{p} = 1$ for all $n$ so the term does not go to zero, and the series fails to converge in $Z_{p}$ . Rubric: full credit for the convergence criterion $∣ a_{n} ∣_{p} \to 0$ plus correct application to $n! p^{n}$ and to a constant-term series.

Advanced results Master

Theorem (Ostrowski 1916; Acta Math. 41). Every absolute value on $Q$ that is not the identity-on-non-zero-elements function is equivalent to either the standard Euclidean absolute value $∣ \cdot ∣_{\infty}$ or to one of the $p$ -adic absolute values $∣ \cdot ∣_{p}$ , one per rational prime $p$ . Two absolute values are equivalent when they induce the same topology, equivalently when one is a positive power of the other.

The proof splits on whether the absolute value $∥ \cdot ∥$ satisfies $∥ n ∥ \leq 1$ for every $n \in Z$ (the non-Archimedean case) or $∥ n ∥ > 1$ for some $n \in Z$ (the Archimedean case). In the non-Archimedean case, the set ${n \in Z : ∥ n ∥ < 1}$ is a non-zero prime ideal of $Z$ (closed under sums by the strong triangle inequality, closed under multiplication by integers, and not the full $Z$ since $∥1∥ = 1$ ). This prime ideal is $p Z$ for some prime $p$ , and one identifies $∥ \cdot ∥$ with $∣ \cdot ∣_{p}^{s}$ for some $s > 0$ . In the Archimedean case, a direct base- $n$ expansion argument compares $∥ \cdot ∥$ to $∣ \cdot ∣_{\infty}^{s}$ and forces them to coincide up to a power. The full argument is given in the Full proof set.

Theorem (compactness and topology). The ring $Z_{p}$ is compact, Hausdorff, totally disconnected, and metrisable. The field $Q_{p}$ is locally compact, Hausdorff, totally disconnected, complete, and metrisable. The topology on $Q_{p}$ is generated by the basis of open balls $B (a, p^{- n}) = a + p^{n} Z_{p}$ for $a \in Q_{p}, n \in Z$ , each of which is simultaneously open and closed.

These topological properties identify $Q_{p}$ as a local field in the sense of Cassels, the prototype of a non-archimedean local field of characteristic zero. The compactness of $Z_{p}$ is the key technical fact: it underlies the existence of a Haar measure on $Q_{p}$ , the finite-dimensionality of every quotient $Z_{p} / p^{n} Z_{p}$ , and the profinite structure $Z_{p} = lim_{n} Z / p^{n} Z$ .

Theorem ( $p$ -adic expansion). Every $x \in Q_{p}^{\times}$ has a unique representation as a convergent series $$ x = \sum_{n \geq n_0} a_n p^n, \qquad a_n \in {0, 1, \ldots, p - 1}, \quad a_{n_0} \neq 0, $$ where $n_{0} = v_{p} (x) \in Z$ . The element $x$ lies in $Z_{p}$ if and only if $n_{0} \geq 0$ , and in $Z_{p}^{\times}$ if and only if $n_{0} = 0$ .

The digit set ${0, 1, \dots, p - 1}$ comes from a chosen set of coset representatives for $F_{p} = Z_{p} / p Z_{p}$ ; the canonical choice is the inclusion of ${0, \dots, p - 1}$ into $Z$ and thence into $Z_{p}$ . The map $x \mapsto (a_{n})_{n \geq n_{0}}$ is a homeomorphism (but not a ring homomorphism) from $Q_{p}$ to the space of eventually- $p$ -bounded sequences, a fact useful for measure-theoretic and analytic constructions.

Theorem (residue field and Teichmüller lift). The reduction map $Z_{p} ↠ F_{p}$ admits a unique multiplicative section $ω : F_{p}^{\times} \to Z_{p}^{\times}$ , the Teichmüller character, characterised by $ω (\overset{a}{ˉ})^{p} = ω (\overset{a}{ˉ})$ for every $\overset{a}{ˉ} \in F_{p}^{\times}$ . The image is the group of $(p - 1)$ -th roots of unity in $Z_{p}$ .

Construction: given $\overset{a}{ˉ} \in F_{p}^{\times}$ , choose any lift $a \in Z_{p}^{\times}$ and form the sequence $a, a^{p}, a^{p^{2}}, \dots$ , which is Cauchy by Fermat's little theorem (each step changes $a$ by a $p$ -adic small correction). The limit $ω (\overset{a}{ˉ}) := lim_{n} a^{p^{n}}$ exists in $Z_{p}^{\times}$ , equals a $(p - 1)$ -th root of unity, and is independent of the chosen lift. The Teichmüller character provides the multiplicative splitting $Z_{p}^{\times} ≅ μ_{p - 1} \times (1 + p Z_{p})$ .

Theorem (unramified extensions, prelude to $Q_{p}^{unr}$ ). For every $n \geq 1$ , there is a unique unramified extension $Q_{p^{n}}$ of $Q_{p}$ of degree $n$ , obtained by adjoining a primitive $(p^{n} - 1)$ -th root of unity. The Galois group is cyclic of order $n$ , generated by the arithmetic Frobenius $Frob_{p}$ acting on the residue field $F_{p^{n}}$ by $\overset{a}{ˉ} \mapsto \overset{a}{ˉ}^{p}$ . The compositum $Q_{p}^{unr} = ⋃_{n} Q_{p^{n}}$ is the maximal unramified extension of $Q_{p}$ , with Galois group $Gal (Q_{p}^{unr} / Q_{p}) ≅ Z$ .

The unramified-extension correspondence places the finite extensions of $F_{p}$ in bijection with the unramified extensions of $Q_{p}$ , with the bijection $F_{p^{n}} \leftrightarrow Q_{p^{n}}$ commuting with Galois actions on both sides. This is the local-field analogue of the bijection between étale algebras over a residue field and unramified covers; in the function-field setting it corresponds to the bijection between étale covers of a smooth affine variety and finite extensions of its fraction field.

Theorem (Hensel's lemma teaser — proved in the next unit). Let $f \in Z_{p} [x]$ and suppose $\overset{a}{ˉ} \in F_{p}$ is a simple root of $\overset{ˉ}{f} \in F_{p} [x]$ (the reduction modulo $p$ ), meaning $\overset{ˉ}{f} (\overset{a}{ˉ}) = 0$ and $\overset{ˉ}{f}^{'} (\overset{a}{ˉ}) \neq = 0$ . There exists a unique $a \in Z_{p}$ with $f (a) = 0$ and $a \equiv \overset{a}{ˉ} (mod p)$ .

This is the foundational lifting principle for $p$ -adic root-finding. The proof is Newton iteration $a_{n + 1} = a_{n} - f (a_{n}) / f^{'} (a_{n})$ in $Z_{p}$ , which converges quadratically thanks to the $p$ -adic metric. The lemma underlies the structure theory of $Q_{p}$ : every quadratic extension corresponds to a square class in $Q_{p}^{\times} / (Q_{p}^{\times})^{2}$ , every $p$ -th power non-residue determines a totally ramified extension, and every unramified extension is generated by a Teichmüller lift. Hensel's lemma gets a full unit in 21.02.04.

Theorem (product formula on $Q^{\times}$ ). For every $x \in Q^{\times}$ , $$ \prod_{v} |x|v = 1, $$ *where the product is over all places $v$ of $Q$ — meaning the Archimedean place $|\cdot|\infty $an d t h e n o n - a r c him e d e an pl a ces$ |\cdot|_p $, o n e p er p r im e$ p$.*

The product is finite for every $x$ since $∣ x ∣_{p} = 1$ for all but finitely many primes (those not dividing the numerator or denominator of $x$ ). The formula expresses a global constraint on the local absolute values: each rational number is "balanced" across the places. This identity is the seed of the adelic perspective, where $Q$ embeds diagonally in the restricted product $A_{Q} = R \times \prod_{p}^{'} Q_{p}$ .

Theorem (the additive structure of $Z_{p}$ ). As an abelian topological group, $Z_{p}$ is the pro- $p$ -completion of $Z$ : it is the inverse limit of the finite cyclic groups $Z / p^{n} Z$ along the natural reduction maps. It contains $Z$ as a dense subgroup, and its proper closed subgroups are exactly the $p^{n} Z_{p}$ for $n \geq 0$ , the zero subgroup ${0}$ , and the limit case.

This structure makes $Z_{p}$ the prototype of a pro- $p$ -group: a topological group whose every open normal subgroup has $p$ -power index. Galois groups of $Z_{p}$ -extensions of number fields, $p$ -adic Lie groups, and Iwasawa-theoretic objects all sit inside the same framework.

Synthesis. The two constructions of $Z_{p}$ — inverse limit and metric completion — are the foundational reason every subsequent structure of $p$ -adic number theory makes sense. The central insight is that the metric topology on $Q$ generated by $∣ \cdot ∣_{p}$ is exactly the topology obtained by viewing the inclusions $Z \subset Z / p^{n} Z$ as a filtration and completing accordingly. Putting these together, the field $Q_{p}$ is the unique complete field containing $Q$ in which $p \to 0$ as $n \to \infty$ in $p^{n}$ , and this is exactly the local field over which Hensel's lemma, the Teichmüller character, the unramified-extension correspondence, and the residue field $F_{p}$ all live coherently. The bridge between the algebraic picture (inverse limits, Galois theory of $Q_{p}^{unr}$ ) and the analytic picture (Cauchy sequences, $p$ -adic series convergence) is exactly the equivalence-of-constructions theorem proved above. The pattern identifies the prime $p$ as a place of $Q$ , on equal footing with the Archimedean place at infinity, and generalises through Ostrowski's classification to the statement that there are no other places to find.

Ostrowski's theorem is the structural fact that closes the question of which absolute values exist on $Q$ : the answer is one Archimedean and one per prime, and no others. This determines the places of $Q$ once and for all, and the completion at each place produces either $R$ (at infinity) or $Q_{p}$ (at the prime $p$ ). The product formula $\prod_{v} ∣ x ∣_{v} = 1$ binds the local pieces into a single global identity, the seed of the adelic framework in which $Q$ embeds diagonally in $A_{Q}$ . The pattern identifies number-theoretic completion with topological completion in a way that generalises immediately to every number field $K$ , where the places of $K$ are still classifiable and each completion $K_{v}$ is a local field. This unification of arithmetic and analysis under one absolute-value framework is exactly the bridge that appears again in 21.03.03 Dedekind, Hecke, and Artin $L$ -functions, where the Euler product over places of $K$ encodes the same local-global structure analytically.

Full proof set Master

Proposition (Ostrowski's theorem; Ostrowski 1916 ^{[source pending]}). Every absolute value $∥ \cdot ∥$ on $Q$ that is not the identity-on-non-zero-elements function is equivalent to either $∣ \cdot ∣_{\infty}$ or to $∣ \cdot ∣_{p}$ for a unique rational prime $p$ .

Proof. Two absolute values $∥ \cdot ∥_{1}$ and $∥ \cdot ∥_{2}$ on $Q$ are equivalent when $∥ x ∥_{1} < 1 ⟺ ∥ x ∥_{2} < 1$ for every $x$ , or equivalently when there exists $s > 0$ such that $∥ x ∥_{2} = ∥ x ∥_{1}^{s}$ for every $x$ . The non-vanishing of $∥ n ∥$ for $n \in Z ∖ {0}$ is automatic from positivity. The key splitting is whether $∥ n ∥ \leq 1$ for every $n \in Z$ .

Case 1: non-Archimedean, $∥ n ∥ \leq 1$ for every $n \in Z$ . The strong triangle inequality $∥ x + y ∥ \leq max (∥ x ∥, ∥ y ∥)$ holds automatically here, by the following argument: for any positive integer $N$ , the binomial expansion gives $$ |x + y|^N = |(x + y)^N| = \left|\sum_{k=0}^N \binom{N}{k} x^k y^{N-k}\right| \leq \sum_{k=0}^N \left|\binom{N}{k}\right| |x|^k |y|^{N-k} \leq (N + 1) \max(|x|, |y|)^N, $$ using $∥ (k N) ∥ \leq 1$ since binomial coefficients are integers. Taking $N$ -th roots and letting $N \to \infty$ gives $∥ x + y ∥ \leq max (∥ x ∥, ∥ y ∥)$ . So $∥ \cdot ∥$ is non-Archimedean.

The set $I = {n \in Z : ∥ n ∥ < 1}$ is non-empty (the absolute value is not the identity-on-non-zero-elements function, and $∥1∥ = 1$ , so the strict inequality must occur for some $n$ ). It is an ideal: closed under sums by the strong triangle inequality, closed under multiplication by any integer $m$ via $∥ mn ∥ = ∥ m ∥∥ n ∥ \leq ∥ n ∥ < 1$ . Since $Z$ is a principal ideal domain, $I = (p)$ for some integer $p \geq 2$ . The ideal $I$ is prime: if $ab \in I$ then $∥ a ∥∥ b ∥ = ∥ ab ∥ < 1$ , forcing $∥ a ∥ < 1$ or $∥ b ∥ < 1$ , that is $a \in I$ or $b \in I$ . So $p$ is a prime number.

Now $∥ p ∥ < 1$ , set $c = ∥ p ∥ \in (0, 1)$ , and define $s = - lo g_{p} c$ so that $∥ p ∥ = p^{- s} = ∣ p ∣_{p}^{s}$ . For any integer $n$ , write $n = p^{v} m$ with $p ∤ m$ . Then $m \in / I$ , so $∥ m ∥ = 1$ (since $∥ m ∥ \leq 1$ and $∥ m ∥ < 1$ would put $m \in I$ ), and $∥ n ∥ = ∥ p ∥^{v} = p^{- s v} = ∣ n ∣_{p}^{s}$ . For rationals $n / d$ with $d \neq = 0$ , $∥ n / d ∥ = ∥ n ∥/∥ d ∥ = ∣ n ∣_{p}^{s} /∣ d ∣_{p}^{s} = ∣ n / d ∣_{p}^{s}$ . So $∥ \cdot ∥ = ∣ \cdot ∣_{p}^{s}$ , equivalent to $∣ \cdot ∣_{p}$ .

Case 2: Archimedean, $∥ n ∥ > 1$ for some $n \in Z$ . Let $n_{0} \geq 2$ be the smallest positive integer with $∥ n_{0} ∥ > 1$ , and set $α = lo g_{n_{0}} ∥ n_{0} ∥ > 0$ . The claim is that $∥ n ∥ = n^{α}$ for every positive integer $n$ , hence $∥ \cdot ∥ = ∣ \cdot ∣_{\infty}^{α}$ on $Q^{\times}$ .

For positive integer $n$ , write $n$ in base $n_{0}$ : $n = c_{0} + c_{1} n_{0} + \dots + c_{k} n_{0}^{k}$ with $0 \leq c_{i} < n_{0}$ and $c_{k} \neq = 0$ , so $n_{0}^{k} \leq n < n_{0}^{k + 1}$ . Then $$ |n| \leq \sum_{i=0}^k |c_i| |n_0|^i \leq \sum_{i=0}^k 1 \cdot n_0^{\alpha i} \leq (k + 1) n_0^{\alpha k} \leq C(\log_{n_0} n + 1) n^\alpha, $$ where $∥ c_{i} ∥ \leq 1$ holds because $c_{i} < n_{0}$ and $n_{0}$ is the smallest integer with $∥ n_{0} ∥ > 1$ . Take $n = N^{t}$ for a positive integer $N$ and large $t$ : $$ |N|^t = |N^t| \leq C (t \log_{n_0} N + 1) N^{\alpha t}. $$ Taking $t$ -th roots and letting $t \to \infty$ gives $∥ N ∥ \leq N^{α}$ .

For the reverse inequality: with $n$ in base $n_{0}$ as above, $n_{0}^{k + 1} > n$ , so by what we just showed, $∥ n_{0}^{k + 1} ∥ = n_{0}^{α (k + 1)} \geq ∥ n ∥ + ∥ n_{0}^{k + 1} - n ∥ \geq ∥ n ∥$ , and $∥ n_{0}^{k + 1} - n ∥ \leq (n_{0}^{k + 1} - n)^{α} \leq (n_{0}^{k + 1} - n_{0}^{k})^{α}$ since $n \geq n_{0}^{k}$ . So $$ |n| \geq n_0^{\alpha(k+1)} - (n_0^{k+1} - n_0^k)^\alpha = n_0^{\alpha(k+1)} \left(1 - (1 - n_0^{-1})^\alpha\right). $$ Set $C^{'} = 1 - (1 - n_{0}^{- 1})^{α} > 0$ . Then $∥ n ∥ \geq C^{'} n_{0}^{α (k + 1)} \geq C^{'} n^{α}$ since $n_{0}^{k + 1} > n$ . Taking $n = N^{t}$ and $t$ -th roots as before forces $∥ N ∥ \geq N^{α}$ . Combining, $∥ N ∥ = N^{α}$ for every positive integer $N$ , hence $∥ \cdot ∥ = ∣ \cdot ∣_{\infty}^{α}$ . $□$

Proposition ( $p$ -adic expansion uniqueness). Every $x \in Z_{p}$ has a unique representation $x = \sum_{n \geq 0} a_{n} p^{n}$ with $a_{n} \in {0, 1, \dots, p - 1}$ . The corresponding statement holds for $Q_{p}^{\times}$ with sums starting from $n = v_{p} (x)$ .

Proof. Existence: given $x \in Z_{p}$ , define $a_{0} \in {0, \dots, p - 1}$ as the unique integer in this range with $x \equiv a_{0} (mod p)$ . Set $x_{1} = (x - a_{0}) / p \in Z_{p}$ , and define $a_{1} = x_{1} mod p$ , $x_{2} = (x_{1} - a_{1}) / p$ , and so on inductively. The partial sums $S_{N} = \sum_{n = 0}^{N - 1} a_{n} p^{n}$ satisfy $x - S_{N} = p^{N} x_{N} \in p^{N} Z_{p}$ , so $∣ x - S_{N} ∣_{p} \leq p^{- N} \to 0$ and $S_{N} \to x$ in $Z_{p}$ . The series $\sum a_{n} p^{n}$ converges to $x$ .

Uniqueness: if $x = \sum a_{n} p^{n} = \sum b_{n} p^{n}$ with both digit sequences in ${0, \dots, p - 1}$ , subtract to get $\sum (a_{n} - b_{n}) p^{n} = 0$ in $Z_{p}$ . Reduce modulo $p$ : $a_{0} \equiv b_{0} (mod p)$ , forcing $a_{0} = b_{0}$ since both are in ${0, \dots, p - 1}$ . Divide by $p$ and repeat to get $a_{n} = b_{n}$ for every $n$ .

For $x \in Q_{p}^{\times}$ with $v_{p} (x) = n_{0}$ , write $x = p^{n_{0}} u$ with $u \in Z_{p}^{\times}$ and apply the existence-uniqueness to $u$ with the first digit $a_{n_{0}} = u mod p$ non-zero. $□$

Proposition (compactness of $Z_{p}$ ). $Z_{p}$ is a compact topological space.

Proof. Use Construction A: $Z_{p} = lim_{n} Z / p^{n} Z$ , viewed as a closed subspace of $\prod_{n \geq 1} Z / p^{n} Z$ with each factor finite discrete. The product is compact by Tychonoff's theorem. The inverse-limit subspace is defined by the compatibility relations $x_{n + 1} \equiv x_{n} (mod p^{n})$ for each $n$ , each of which cuts out a closed subset (it is the equalizer of two continuous maps from $\prod_{k} Z / p^{k} Z$ to $Z / p^{n} Z$ ). A closed subspace of a compact space is compact. $□$

Proposition (total disconnectedness). The only connected subsets of $Q_{p}$ are singletons.

Proof. This was the content of Exercise 7. Repeat for the record: suppose $C \subseteq Q_{p}$ is connected with two distinct points $x \neq = y$ . Set $r = ∣ x - y ∣_{p} > 0$ , and form the open ball $B (x, r)$ . In an ultrametric, every open ball is closed: if $z_{n} \in B (x, r)$ with $z_{n} \to z$ , then for large $n$ we have $∣ z - z_{n} ∣_{p} < r$ and $∣ z_{n} - x ∣_{p} < r$ , so $∣ z - x ∣_{p} \leq max (∣ z - z_{n} ∣_{p}, ∣ z_{n} - x ∣_{p}) < r$ , putting $z \in B (x, r)$ . Now $C = (C \cap B (x, r)) ⊔ (C ∖ B (x, r))$ , both pieces non-empty (first contains $x$ , second contains $y$ ) and both open in the subspace topology. This contradicts connectedness. $□$

Proposition (local compactness of $Q_{p}$ ). $Q_{p}$ is locally compact.

Proof. Every point $x \in Q_{p}$ has the compact neighbourhood $x + Z_{p}$ , since translation is a homeomorphism and $Z_{p}$ is compact. The collection of compact neighbourhoods ${x + p^{n} Z_{p}}_{n \in Z}$ is a neighbourhood basis at $x$ . $□$

Proposition (residue-field identification). $Z_{p} / p Z_{p} ≅ F_{p}$ as rings.

Proof. From the inverse-limit description, the projection $Z_{p} \to Z / p Z$ onto the first component is a surjective ring homomorphism. Its kernel is ${(x_{n}) : x_{1} = 0} = {(x_{n}) : x_{n} \in p Z / p^{n} Z for every n}$ . The latter is exactly $p Z_{p}$ , the principal ideal generated by $p$ in $Z_{p}$ . So $Z_{p} / p Z_{p} ≅ Z / p Z = F_{p}$ . $□$

Proposition (Teichmüller existence). For every $\overset{a}{ˉ} \in F_{p}^{\times}$ , the sequence $a, a^{p}, a^{p^{2}}, \dots$ in $Z_{p}^{\times}$ (for any lift $a$ of $\overset{a}{ˉ}$ ) is Cauchy and converges to a unique $ω (\overset{a}{ˉ}) \in Z_{p}^{\times}$ with $ω (\overset{a}{ˉ})^{p} = ω (\overset{a}{ˉ})$ .

Proof. Lift $\overset{a}{ˉ}$ to any $a \in Z_{p}^{\times}$ . By Fermat's little theorem $\overset{a}{ˉ}^{p} = \overset{a}{ˉ}$ in $F_{p}^{\times}$ , so $a^{p} \equiv a (mod p)$ , that is $a^{p} - a \in p Z_{p}$ . Then $$ a^{p^{n+1}} - a^{p^n} = a^{p^n}(a^{p^n (p - 1)} - 1) = a^{p^n} \prod_{\zeta^{p - 1} = 1}(a^{p^n} - \zeta) $$ working in a finite-degree extension; the key estimate is $v_{p} (a^{p^{n + 1}} - a^{p^{n}}) \geq n + 1$ , proved by induction on $n$ using the lift of Frobenius: assuming $a^{p^{n}} - a^{p^{n - 1}} \in p^{n} Z_{p}$ , raise both sides to the $p$ -th power and use the binomial expansion, where every cross-term picks up a factor of $p$ from the binomial coefficient and a factor of $p^{n}$ from the base inductive estimate, giving total valuation at least $n + 1$ . So $(a^{p^{n}})_{n}$ is Cauchy and converges to a limit $ω = ω (\overset{a}{ˉ}) \in Z_{p}^{\times}$ . Taking limits in the relation $a^{p^{n + 1}} = (a^{p^{n}})^{p}$ gives $ω = ω^{p}$ , so $ω^{p - 1} = 1$ and $ω$ is a $(p - 1)$ -th root of unity. Uniqueness: any two $(p - 1)$ -th roots of unity in $Z_{p}^{\times}$ congruent modulo $p$ are equal, since their difference is a $p$ -adic unit times a difference of roots of unity which is itself a unit or zero. $□$

Proposition ( $Q_{p}$ is complete). Every Cauchy sequence in $Q_{p}$ converges in $Q_{p}$ .

Proof. A Cauchy sequence $(x_{k})$ in $Q_{p}$ has bounded absolute values: $∣ x_{k} ∣_{p} \leq p^{M}$ for some $M$ and all $k$ . Multiply by $p^{M}$ to get a Cauchy sequence in $Z_{p}$ , which converges by compactness (or directly by the inverse-limit description: the residues modulo $p^{n}$ stabilise for each $n$ , and the limit is the coherent system of stabilised residues). Divide by $p^{M}$ to recover the limit in $Q_{p}$ . $□$

Connections Master

Riemann zeta function 21.03.01. The Euler product $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ expresses the global Riemann zeta function as a product over primes of local factors, each of which is built from the arithmetic of $Q_{p}$ . The pole at $s = 1$ in the global zeta function reflects the divergence of $\sum 1/ n$ in the ordinary norm; the absence of poles at finite primes reflects the contrast that $\sum p^{n}$ converges in $Z_{p}$ . The $p$ -adic numbers are exactly the local field at which the local Euler factor $(1 - p^{- s})^{- 1}$ is computed.
Hensel's lemma and unramified extensions 21.02.04. The immediate successor unit: Hensel's lemma is the lifting principle for simple roots of polynomials from $F_{p}$ to $Z_{p}$ , and the Teichmüller lift is its canonical application. Unramified extensions $Q_{p^{n}}$ are generated by Hensel-lifted roots of irreducible polynomials in $F_{p} [x]$ , and they parametrise the étale algebra structure over $Q_{p}$ . The unramified-extension theory built on top of $Q_{p}$ is the local input to global Galois theory of number fields.
$ℓ$ -adic Galois representations 21.05.01. The Tate module $T_{ℓ} E = lim_{n} E [ℓ^{n}]$ of an elliptic curve $E$ is constructed as an inverse limit exactly parallel to Construction A of $Z_{p}$ , with $Z / p^{n} Z$ replaced by the $ℓ^{n}$ -torsion group scheme. The Galois action on $T_{ℓ} E$ gives a continuous representation $ρ_{E, ℓ} : G_{Q} \to GL_{2} (Z_{ℓ})$ , and the topology on the codomain is the $ℓ$ -adic topology developed here. Every $ℓ$ -adic Galois representation is a continuous map into a $GL_{n} (Z_{ℓ})$ , so the foundations of $Z_{p}$ -theory directly underpin the entire $ℓ$ -adic representation framework.
$Z_{p}$ -extensions and Iwasawa theory 21.07.01. A $Z_{p}$ -extension of a number field $K$ is a Galois extension with $Gal (K_{\infty} / K) ≅ Z_{p}$ as a topological group; the cyclotomic $Z_{p}$ -extension of $Q$ is built from the $p^{n}$ -th roots of unity. The Iwasawa algebra $Λ = Z_{p} [[Γ]] ≅ Z_{p} [[T]]$ is a completed group ring built on $Z_{p}$ , and the structure theorem for finitely generated torsion $Λ$ -modules — central to Iwasawa theory — is a structure-theoretic statement about $Z_{p} [[T]]$ that ultimately rests on the topological-ring structure of $Z_{p}$ developed here.
$p$ -adic $L$ -functions 21.07.02. The Kubota-Leopoldt $p$ -adic $L$ -function $L_{p} (s, χ) : Z_{p} \to C_{p}$ interpolates classical Dirichlet $L$ -values at negative integers, and its existence depends on the topology of $Z_{p}$ as a compact totally-disconnected space — the values of $L_{p}$ at integer points form a dense subset, and the continuity required to interpolate them is $p$ -adic continuity on $Z_{p}$ . The Amice transform identifies continuous $C_{p}$ -valued functions on $Z_{p}$ with elements of the Iwasawa algebra, transporting the analytic interpolation into an algebraic object.
Absolute value and triangle inequality 00.01.02. The foundational unit on absolute values introduces the four signature properties — positivity, multiplicativity, triangle inequality, reverse triangle — and discusses the ultrametric strengthening as a Master-tier remark. The $p$ -adic absolute value developed here is the prototype of an ultrametric, satisfying the strong triangle inequality $∣ x + y ∣_{p} \leq max (∣ x ∣_{p}, ∣ y ∣_{p})$ instead of the ordinary $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ . Ostrowski's theorem closes the question of which absolute values on $Q$ exist, classifying every absolute value other than the identity-on-non-zero-elements function as either Euclidean or $p$ -adic.

Historical & philosophical context Master

Kurt Hensel introduced the $p$ -adic numbers in his 1897 Jahresbericht der Deutschen Mathematiker-Vereinigung paper Über eine neue Begründung der Theorie der algebraischen Zahlen ^{[source pending]} by analogy with the function-field case familiar from algebraic geometry. For a smooth complex curve $X$ and a closed point $P \in X$ , a meromorphic function near $P$ admits a Laurent expansion in a local parameter, and the local ring at $P$ is the formal power series ring $C [[t]]$ with fraction field $C ((t))$ . Hensel proposed that a rational prime $p$ should play the role of a "point on the spectrum of $Z$ ", with a local ring of $p$ -adic integers $Z_{p}$ and a fraction field $Q_{p}$ analogous to $C ((t))$ . His original construction was the formal power series $\sum_{n \geq 0} a_{n} p^{n}$ with digits $a_{n} \in {0, \dots, p - 1}$ and carrying rules borrowed from base- $p$ arithmetic. The 1908 Theorie der algebraischen Zahlen (Teubner) expanded the framework into a foundation for algebraic number theory and introduced Hensel's lemma explicitly as the lifting principle for roots.

The metric-completion construction came later, as part of the general theory of valuations developed by Wolfgang Krull in the 1930s and synthesised in Bourbaki's Algèbre commutative (Ch. VI, 1964). The inverse-limit picture, also implicit in Hensel's original work, became standard with the rise of category theory and was given clean expositions in Serre's 1962 Corps locaux (Hermann) and 1970 Cours d'arithmétique ^{[source pending]}. The two-construction equivalence is the bridge between the analytic origin (Cauchy sequences, completion) and the algebraic origin (inverse limit of finite quotients).

Alexander Ostrowski's 1916 Acta Mathematica paper Über einige Lösungen der Funktionalgleichung $φ (x) φ (y) = φ (x y)$ ^{[source pending]} classified all multiplicative absolute values on $Q$ into the Euclidean and the $p$ -adic families. The result closed an open question that Hensel had raised implicitly: there are no exotic completions of $Q$ beyond $R$ and the $Q_{p}$ . The theorem became the foundation for the adelic perspective, in which $Q$ embeds diagonally in the restricted product $A_{Q} = R \times \prod_{p}^{'} Q_{p}$ over all completions, and number-theoretic questions about $Q$ admit a local-global decomposition into questions at each place.

Émile Artin's 1931 lectures at Notre Dame (published 1959 as Algebraic Numbers and Algebraic Functions) and Helmut Hasse's 1923-1926 papers on the local-global principle (the Hasse-Minkowski theorem for quadratic forms over $Q$ being a prototype) consolidated the framework. André Weil's 1967 Basic Number Theory (Grundlehren 144) cast the entire theory in the language of locally compact topological rings and Haar measure, placing $Q_{p}$ as the prototype of a non-archimedean local field on equal footing with $R$ as the prototype of an archimedean local field. The Langlands programme, beginning with Robert Langlands's 1967 letter to Weil and developed across the 1970s, took the next step by attaching automorphic representations to every place simultaneously.

Bibliography Master

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}

@book{Neukirch1999,
  author    = {Neukirch, J{\"u}rgen},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {322},
  year      = {1999}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yu. I. and Panchishkin, A. A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2},
  year      = {2005}
}

@book{Schikhof1984,
  author    = {Schikhof, W. H.},
  title     = {Ultrametric Calculus},
  publisher = {Cambridge University Press},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {4},
  year      = {1984}
}

@book{Robert2000,
  author    = {Robert, Alain M.},
  title     = {A Course in p-adic Analysis},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {198},
  year      = {2000}
}

@book{Weil1967,
  author    = {Weil, Andr{\'e}},
  title     = {Basic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {144},
  year      = {1967}
}

Prerequisites

00.01.02
02.01.05
02.03.02
01.02.01

Tier anchors

beginner: Gouvêa *p-adic Numbers: An Introduction* (Springer UTX, 2nd ed. 1997) Ch. 1-3 informal opening on the size $|n|_p$ and the new arithmetic it produces; Koblitz *p-adic Numbers, p-adic Analysis, and Zeta-Functions* (GTM 58, 2nd ed. 1984) Ch. 1 §§1-3
intermediate: Serre *A Course in Arithmetic* (Springer GTM 7, 1973) Ch. 2 §§1-3 (inverse-limit construction, $p$-adic valuation, $\mathbb{Q}_p$ as completion); Koblitz *p-adic Numbers, p-adic Analysis, and Zeta-Functions* Ch. 1-3; Gouvêa Ch. 3-5
master: Hensel 1897 *Jahresbericht DMV* 6, 83-88 (originator: $p$-adic numbers as formal power series in $p$); Hensel 1908 *Theorie der algebraischen Zahlen* (Teubner, Leipzig); Ostrowski 1916 *Acta Math.* 41, 271-284 (classification of absolute values on $\mathbb{Q}$); Serre *A Course in Arithmetic* Ch. 2; Cassels *Local Fields* (LMS Student Texts 3, 1986) Ch. 1-4; Neukirch *Algebraic Number Theory* (Grundlehren 322, 1999) Ch. II §§1-5; Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 1 §§1-3; Schikhof *Ultrametric Calculus* (Cambridge Studies 4, 1984) Ch. 1; Bachman *Introduction to p-adic Numbers and Valuation Theory* (Academic Press 1964); Robert *A Course in p-adic Analysis* (GTM 198, 2000) Ch. 1-2

References

Hensel, K. — Über eine neue Begründung der Theorie der algebraischen Zahlen · *Jahresbericht der Deutschen Mathematiker-Vereinigung* 6 (1897), 83-88. Originator paper for the $p$-adic numbers: Hensel introduced $\mathbb{Q}_p$ as the analogue of formal power series for primes, by analogy with the function-field case where a point on a curve gives a Laurent expansion. The construction was via formal Laurent series in $p$ with digits in $\{0, \ldots, p - 1\}$ and termwise carrying, predating both the completion picture and the inverse-limit picture.
Hensel, K. — Theorie der algebraischen Zahlen · Teubner, Leipzig (1908). The full monograph treatment of $p$-adic numbers as a foundation for algebraic number theory. Hensel's lemma appears explicitly here as the lifting principle for roots of polynomials over $\mathbb{Z}_p$ from residue-field roots over $\mathbb{F}_p$, predicated on the simple-root hypothesis.
Ostrowski, A. — Über einige Lösungen der Funktionalgleichung $\varphi(x) \varphi(y) = \varphi(xy)$ · *Acta Mathematica* 41 (1916), 271-284. The classification theorem: every non-identity multiplicative absolute value on $\mathbb{Q}$ is equivalent to either the standard Euclidean absolute value or one of the $p$-adic absolute values, one per rational prime. The proof exploits the multiplicativity to reduce to the value on integers, then splits on whether the absolute value exceeds $1$ on some integer (Archimedean case) or not (non-Archimedean case).
Serre, J.-P. — A Course in Arithmetic · Springer Graduate Texts in Mathematics 7 (English translation 1973 of *Cours d'arithmétique*, PUF 1970), Ch. 2. The cleanest modern presentation of the two equivalent constructions of $\mathbb{Z}_p$ — inverse limit $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ and metric completion of $\mathbb{Z}$ under $|\cdot|_p$ — together with the $p$-adic expansion and Ostrowski's theorem.
Koblitz, N. — p-adic Numbers, p-adic Analysis, and Zeta-Functions · Springer Graduate Texts in Mathematics 58 (2nd ed. 1984), Ch. 1-3. A pedagogical complement to Serre with explicit examples, the bridge to $p$-adic analysis (continuous functions on $\mathbb{Z}_p$, the Mahler basis), and the seed of $p$-adic $L$-function theory.
Gouvêa, F. Q. — p-adic Numbers: An Introduction · Springer Universitext (2nd ed. 1997), Ch. 1-5. The most accessible textbook introduction: motivates the $p$-adic absolute value by analogy with the function-field case, computes worked examples in detail, and proves Ostrowski's theorem explicitly with a model-Archimedean and model-non-Archimedean split.
Cassels, J. W. S. — Local Fields · Cambridge London Mathematical Society Student Texts 3 (1986), Ch. 1-4. The local-field framework in which $\mathbb{Q}_p$ is the prototype: complete discretely valued fields with finite residue field, their unramified extensions parametrised by extensions of the residue field $\mathbb{F}_p$, and the structure theorem for finite extensions of $\mathbb{Q}_p$.
Neukirch, J. — Algebraic Number Theory · Springer Grundlehren 322 (1999), Ch. II §§1-5. The arithmetic-geometric perspective: $\mathbb{Q}_p$ as the completion of $\mathbb{Q}$ at the prime $p$, the product formula $\prod_v |x|_v = 1$ over all places of $\mathbb{Q}$, and the adelic point of view that places $\mathbb{Q}_p$ alongside $\mathbb{R}$ as the local pieces of $\mathbb{A}_\mathbb{Q}$.
Manin, Yu. I. & Panchishkin, A. — Introduction to Modern Number Theory · Springer Encyclopaedia of Mathematical Sciences 49, 2nd edition (2005), Ch. 1 §§1-3. The encyclopaedic survey covering $\mathbb{Z}_p$ both as inverse limit and as completion, Ostrowski's theorem, the topology of $\mathbb{Q}_p$, and the bridge into Hensel's lemma and unramified extensions.
Schikhof, W. H. — Ultrametric Calculus · Cambridge Studies in Advanced Mathematics 4 (1984), Ch. 1. The ultrametric perspective: the strong triangle inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$ that distinguishes the $p$-adic absolute value from the Euclidean one, and the unusual topological consequences (every triangle is isosceles, every open ball is closed, every point of a ball is a centre).
Robert, A. M. — A Course in p-adic Analysis · Springer Graduate Texts in Mathematics 198 (2000), Ch. 1-2. Modern textbook on $p$-adic analysis with full development of $\mathbb{Z}_p$ and $\mathbb{Q}_p$, including the topological structure (compact totally-disconnected for $\mathbb{Z}_p$, locally compact for $\mathbb{Q}_p$), the Haar measure, and the continuous-function theory underpinning $p$-adic $L$-functions.

Estimated time

beginner: 18m
intermediate: 45m
master: 75m