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p -adic number

In mathematics, the p-adic numbers form a field \mathbb{Q}_p, where p is a fixed prime number, constructed as the completion of the rational numbers \mathbb{Q} with respect to the p-adic metric derived from the p-adic valuation.^[1] This valuation measures the highest power of p dividing a rational number, leading to a non-Archimedean absolute value |x|_p = p^{-v_p(x)}, where v_p(x) is the valuation, which satisfies the ultrametric inequality |x + y|_p \leq \max\{|x|_p, |y|_p\}.^[2] The p-adic integers \mathbb{Z}_p are the subring of elements with |x|_p \leq 1, consisting of formal power series \sum_{i=0}^\infty a_i p^i with coefficients a_i \in \{0, 1, \dots, p-1\}.^[1] Introduced by the German mathematician Kurt Hensel in 1897 as a tool for solving polynomial equations modulo powers of p, the p-adic numbers provide an alternative number system to the reals, emphasizing divisibility by p rather than magnitude.^[3] Hensel's lemma, a cornerstone result, guarantees the lifting of solutions from modulo p to the full p-adic integers under certain conditions, facilitating local analysis in algebraic number theory.^[1] Unlike the real numbers, \mathbb{Q}_p is totally disconnected and locally compact, with every element expressible as a Laurent series \sum_{i=n}^\infty a_i p^i for some integer n, allowing infinite expansions to the left in base p.^[2] The p-adic numbers play a pivotal role in modern number theory, underpinning local-global principles such as the Hasse-Minkowski theorem for quadratic forms, which equates global solvability over \mathbb{Q} to local solvability over \mathbb{Q}_p for all p and over \mathbb{R}.^[4] Ostrowski's theorem classifies all non-trivial absolute values on \mathbb{Q}, showing they are either the standard real one or p-adic for some prime p, highlighting the completeness of these systems.^[1] Applications extend to p-adic analysis, where functions like the exponential and logarithm are defined via power series, and to arithmetic geometry, including the study of elliptic curves over \mathbb{Q}_p.^[2]

Introduction

Motivation

p-adic numbers were introduced by the German mathematician Kurt Hensel in 1897, primarily to establish an analogy between power series expansions in complex analysis and expansions of algebraic integers around a prime ideal in number fields, facilitating the study of Diophantine equations modulo primes and their higher powers.^[5] This innovation allowed for a systematic approach to lifting solutions from modulo p to modulo higher powers of p, generalizing classical results such as quadratic reciprocity by reformulating solvability over the integers in terms of local solvability in these completions at each prime. Hensel's construction addressed limitations in traditional algebraic number theory, where global considerations often obscured local behaviors essential for understanding equations like those in reciprocity laws.^[6] The real numbers, as the archimedean completion of the rationals, excel at capturing approximations and continuous phenomena but fail to naturally accommodate congruences, such as determining whether an equation like x^2 \equiv 1 \pmod{p} admits solutions for every prime p and extends consistently to higher powers p^k.^[7] In contrast, p-adic numbers provide a non-archimedean metric that prioritizes divisibility by p, enabling precise handling of such modular conditions through a topology where proximity is measured by shared trailing digits in base-p representations.^[8] This local perspective at each prime p complements the global view of the reals, forming part of the adelic framework in modern number theory, where solutions to Diophantine equations are analyzed componentwise across all places (primes and infinity).^[7] A striking application arises in Fermat's Last Theorem, where proofs for small exponents employ infinite descent; in the p-adic setting, this descent translates to a sequence of approximate solutions converging via continuity to a full p-adic solution, yielding a contradiction if no nontrivial p-adic root exists for the relevant equation.^[9]

Informal description

p-adic numbers provide an alternative way to extend the rational numbers, analogous to how real numbers complete the rationals with respect to absolute value, but using a different metric based on divisibility by powers of a prime p. Intuitively, one can think of p-adic numbers through their representation in base p, where expansions extend infinitely to the left rather than to the right as in decimal expansions for reals. A typical p-adic number appears as \dots d_2 d_1 d_0 . d_{-1} d_{-2} \dots, with each digit d_i ranging from 0 to p-1, allowing for formal power series \sum_{i=k}^\infty d_i p^i for some integer k. This leftward extension captures "negative powers" in a manner that emphasizes agreement in higher powers of p.^[1]^[10] Another perspective arises from constructing p-adic numbers as limits of sequences of rational approximations that become increasingly congruent modulo higher powers of p. For instance, to find a p-adic solution to an equation, one begins with a solution modulo p and iteratively refines it to satisfy the equation modulo p^2, then p^3, and so forth; the p-adic number is the "limit" where these approximations stabilize in this modular sense. This process mirrors solving systems of congruences and highlights how p-adic numbers encode infinite precision in divisibility properties.^[1] The p-adic sense of "closeness" fundamentally differs from the real numbers: two numbers are close if their difference is divisible by a high power of p, making numbers congruent modulo large p^k nearby. For example, 1 and $1 + p are close in the p-adics since their difference p is divisible by p^1, and increasingly so for higher multiples, whereas in the reals they are separated by distance p. This ultrametric property implies that the "strongest" distance dominates, leading to tree-like topologies where balls are nested in a hierarchical fashion based on p-divisibility.^[1] A concrete illustration in the 2-adics occurs with the infinite series $1 + 2 + 4 + 8 + \dots = \dots 1111_2, which equals -1. The partial sum up to $2^{k-1} is $2^k - 1, congruent to -1 modulo $2^k, so as k increases, the approximations converge to -1 in the 2-adic metric; formally, the geometric series sums to \frac{1}{1-2} = -1. This convergence, impossible in the reals, underscores how the p-adic valuation prioritizes higher powers.^[1]

Formal Definitions

As formal power series

The p-adic numbers can be rigorously defined as formal Laurent series over the prime p with coefficients from the finite set \{0, 1, \dots, p-1\}. Specifically, a p-adic number x \in \mathbb{Q}_p is an infinite sum of the form

x = \sum_{n=k}^{\infty} a_n p^n,

where k \in \mathbb{Z} is the lowest index (possibly negative), each coefficient satisfies a_n \in \{0, 1, \dots, p-1\}, and a_k \neq 0 unless x = 0 (in which case all a_n = 0). This representation is unique for every p-adic number, analogous to but extending infinitely to the left the base-p expansions of rational numbers.^[11] Addition and multiplication of two such series are defined componentwise with respect to powers of p, incorporating carry-over terms exactly as in base-p arithmetic: when the sum or product of coefficients in a given power exceeds or equals p, the excess is carried to the next higher power. These operations make \mathbb{Q}_p into a commutative ring with identity, where the additive identity is the zero series and the multiplicative identity is the series with a_0 = 1 and a_n = 0 for all n \neq 0. In fact, \mathbb{Q}_p forms a field under these operations, as every nonzero element has a multiplicative inverse, which can be computed algorithmically via similar series manipulations.^[11] The p-adic valuation v_p(x) of a nonzero p-adic number x is defined as the minimal index n such that a_n \neq 0, with v_p(0) = +\infty by convention. The associated p-adic absolute value is then |x|_p = p^{-v_p(x)}, which satisfies |x|_p = 0 if and only if x = 0. This valuation distinguishes units in \mathbb{Q}_p^\times as those series with v_p(x) = 0 (i.e., a_0 \neq 0).^[11] For an example, consider p = [3](/page/3): the rational number $1/2 has the $3-adic expansion $1/2 = \dots 11112_3 = 2 + 3 + 3^2 + 3^3 + \cdots, where the coefficients are a_0 = 2 and a_n = 1 for all n \geq 1. This series satisfies the equation $2x = 1 in the $3-adics under the defined multiplication, confirming its representation.^[10]

As completion of the rationals

The p-adic numbers can be constructed analytically as the completion of the rational numbers \mathbb{Q} with respect to the p-adic metric, providing a framework that emphasizes limits and convergence in a non-Archimedean topology.^[1] This approach parallels the construction of the real numbers as the completion of \mathbb{Q} under the usual absolute value, but uses a different metric derived from a valuation specific to a fixed prime p.^[12] The foundation is the p-adic valuation v_p on \mathbb{Q}. For a nonzero rational q \in \mathbb{Q}, write q = p^k \cdot (a/b) where a, b \in \mathbb{Z}, p \nmid a, and p \nmid b; then v_p(q) = k. This extends multiplicatively: v_p(q_1 q_2) = v_p(q_1) + v_p(q_2), and v_p(0) = \infty. The valuation satisfies the ultrametric inequality v_p(x + y) \geq \min\{v_p(x), v_p(y)\} for all x, y \in \mathbb{Q}.^[1]^[12] From this valuation arises the p-adic absolute value | \cdot |_p: \mathbb{Q} \to \mathbb{R}_{\geq 0}, defined by |q|_p = p^{-v_p(q)} for q \neq 0 and |0|_p = 0. This induces a metric d_p(x, y) = |x - y|_p on \mathbb{Q}, turning \mathbb{Q} into a metric space. The metric is non-Archimedean, meaning |x + y|_p \leq \max\{|x|_p, |y|_p\}, which implies that triangles are "isosceles" in a strong sense and leads to unusual convergence behaviors compared to the Euclidean metric.^[1]^[12] In this metric space, a sequence (x_n) in \mathbb{Q} is Cauchy if for every \epsilon > 0, there exists N \in \mathbb{N} such that d_p(x_m, x_n) < \epsilon for all m, n \geq N, or equivalently, |x_{n+1} - x_n|_p \to 0 as n \to \infty. However, \mathbb{Q} is not complete under d_p; there exist Cauchy sequences that do not converge within \mathbb{Q}. For example, the sequence defined by partial sums approximating a p-adic limit, such as solving x^2 = a for a quadratic non-residue modulo p, may diverge in \mathbb{Q} but converge in the completion.^[1] The field of p-adic numbers, denoted \mathbb{Q}_p, is the metric completion of \mathbb{Q} with respect to d_p. Formally, \mathbb{Q}_p consists of equivalence classes of Cauchy sequences in \mathbb{Q}, where two sequences (x_n) and (y_n) are equivalent if |x_n - y_n|_p \to 0 as n \to \infty (i.e., they differ by a null sequence converging to 0 in the p-adic sense). Addition and multiplication are defined componentwise on representatives, and the metric extends continuously to \mathbb{Q}_p, making it a complete metric space and a field extending \mathbb{Q}. This construction ensures every Cauchy sequence in \mathbb{Q}_p converges, enabling analytic tools like power series expansions in p-adic analysis.^[1]^[12]

Equivalent formulations

The p-adic integers \mathbb{Z}_p can be defined as the inverse limit \varprojlim_{n} \mathbb{Z}/p^n \mathbb{Z} in the category of rings, where the transition maps are the natural projections modulo p^n for n \geq m.^[13] Elements of this inverse limit consist of threads of compatible residue classes: sequences (a_n)_{n \in \mathbb{N}} with a_n \in \mathbb{Z}/p^n \mathbb{Z} such that a_{n+1} \equiv a_n \pmod{p^n} for all n.^[14] This construction yields a compact Hausdorff topological ring under the inverse limit topology, where the basic open sets are the kernels of the projections \pi_n: \mathbb{Z}_p \to \mathbb{Z}/p^n \mathbb{Z}.^[15] The field of p-adic numbers \mathbb{Q}_p is then obtained as the field of fractions of \mathbb{Z}_p, or equivalently as the localization of \mathbb{Z}_p at the prime ideal (p).^[14] This formulation unifies the algebraic structure, ensuring \mathbb{Q}_p is a locally compact field with respect to the induced topology.^[13] In the residue system approach, elements of \mathbb{Z}_p are identified with compatible systems of residues (a_k \mod p^k)_{k \geq 1}, where a_{k+1} \equiv a_k \pmod{p^k} and typically $0 \leq a_k < p^k for a canonical choice.^[16] This perspective emphasizes the projective nature of the limit, allowing \mathbb{Z}_p to be viewed as the set of all such coherent sequences under componentwise addition and multiplication modulo p^k.^[15] To ensure unique representations, normalization is imposed by restricting the "digits" in the associated p-adic expansion to $0 \leq b_i < p, corresponding to the unique lift where a_k = \sum_{i=0}^{k-1} b_i p^i. The normalized form aligns with the inverse limit by mapping partial sums \sum_{i=0}^{k-1} b_i p^i \mod p^k to the residue a_k.^[13]^[14] All standard definitions of the p-adic numbers yield isomorphic fields: the formal power series construction is isomorphic to the inverse limit via the map sending a series \sum_{i=0}^\infty b_i p^i (with $0 \leq b_i < p) to the sequence of its partial sums modulo p^k, which is bijective and preserves ring operations.^[15] Similarly, the completion of \mathbb{Q} with respect to the p-adic valuation embeds densely into this structure, yielding the same field \mathbb{Q}_p.^[14] These isomorphisms hold for any prime p, confirming the equivalence across analytic, algebraic, and formal perspectives.^[13]

Notation and Expansions

Standard notation

The p-adic valuation v_p(x) of a nonzero rational number x is defined as the highest power of the prime p that divides x, extended to the p-adic numbers, where v_p(0) = +\infty.^[2] The associated p-adic absolute value is given by |x|_p = p^{-v_p(x)} for x \neq 0, and |0|_p = 0; this satisfies the non-Archimedean triangle inequality |x + y|_p \leq \max\{|x|_p, |y|_p\}.^[1]^[17] The ring of p-adic integers is denoted \mathbb{Z}_p = \{ x \in \mathbb{Q}_p : |x|_p \leq 1 \}, consisting of elements with nonnegative valuation, while the field of p-adic numbers is \mathbb{Q}_p, the completion of the rationals with respect to the p-adic metric induced by |\cdot|_p.^[2]^[17] Open balls in this metric are written as B(a, r) = \{ x \in \mathbb{Q}_p : |x - a|_p < r \}, where r > 0; in particular, balls centered at 0 are denoted B(0, r) = \{ x : |x|_p < r \}.^[1] Elements of \mathbb{Q}_p are represented as formal Laurent series x = \sum_{n=v}^\infty a_n p^n, where v = v_p(x) \in \mathbb{Z} \cup \{+\infty\}, the coefficients a_n are integers satisfying $0 \leq a_n < p, and only finitely many negative powers appear.^[2]^[17] A "p-adic decimal point" is often placed before the term of degree 0 to separate the integer and fractional parts, analogous to decimal expansions in the reals.^[1] Throughout the literature, p is conventionally taken to be a prime number, ensuring the non-Archimedean property, though the formal construction extends to any integer base greater than 1; the case p = \infty recovers the archimedean absolute value on the reals, which is unbounded on the integers unlike the p-adic norms.^[2]^[1]^[17]

p-adic expansions of rational numbers

Every rational number q \in \mathbb{Q} embeds uniquely into the field of p-adic numbers \mathbb{Q}_p as a p-adic Laurent series q = \sum_{k = v}^{\infty} c_k p^k, where v = v_p(q) is the p-adic valuation of q, and each coefficient satisfies $0 \leq c_k \leq p-1. This representation is unique except in cases of "terminating" expansions, where an infinite tail of (p-1)'s can be replaced by carrying over to the next digit, analogous to the non-uniqueness of $0.999\ldots = 1 in decimal expansions.^[10]^[18] To compute the p-adic expansion of a rational q = a/b in lowest terms, first compute the valuation v = v_p(a) - v_p(b), which determines the lowest power of p in the series. The expansion then consists of finitely many terms for negative powers if v < 0, followed by the expansion of the p-adic unit u = p^{-v} q. The digits of u are found successively by solving for approximations modulo increasing powers of p: start with u_0 \equiv u \pmod{p}, so c_0 = u_0; then lift to u_{k+1} = u_k + c_{k+1} p^{k+1} where c_{k+1} is chosen such that u_{k+1} \equiv u \pmod{p^{k+2}}, with $0 \leq c_{k+1} \leq p-1. This process, akin to long division in base p, converges in the p-adic topology to u.^[10]^[19] The nature of the expansion depends on the prime factors of the denominator b. If b is coprime to p (after reducing a/b), then v = 0 and the expansion is infinite in the nonnegative powers and eventually periodic, with the period dividing the order of the multiplicative group (\mathbb{Z}/p^m \mathbb{Z})^\times for some m, reflecting the finite structure of units modulo p^m. If p divides b but no other primes do (i.e., b = p^{-v} times a unit), the expansion terminates after the negative powers, with all higher coefficients zero. In general, when b has both p-power and coprime factors, the expansion features finitely many negative powers followed by an eventually periodic sequence in the nonnegative powers.^[10]^[18] A concrete example is the 5-adic expansion of $1/3, where the denominator 3 is coprime to 5, so v_5(1/3) = 0 and the series is eventually periodic with period 2. The successive approximation yields digits c_0 = 2, c_1 = 3, c_2 = 1, and then repeating 3, 1 thereafter:

\frac{1}{3} = \dots + 1 \cdot 5^2 + 3 \cdot 5^3 + 1 \cdot 5^4 + 3 \cdot 5^5 + 1 \cdot 5^6 + \cdots + 3 \cdot 5^1 + 2 \cdot 5^0

in the 5-adic sense (written from higher to lower powers for readability). Verifying the partial sum up to $5^2: $2 + 3 \cdot 5 + 1 \cdot 25 = 2 + 15 + 25 = 42, and $3 \cdot 42 = 126 \equiv 1 \pmod{125}, consistent with the lifting process; higher terms refine the congruence to equality in \mathbb{Q}_5.^[10]^[20]

p-adic Integers

Definition and construction

The p-adic integers, denoted \mathbb{Z}_p, form the ring of integers in the field of p-adic numbers \mathbb{Q}_p, where p is a fixed prime number. They are defined as the closed unit ball in \mathbb{Q}_p with respect to the p-adic absolute value:

\mathbb{Z}_p = \{ x \in \mathbb{Q}_p : |x|_p \leq 1 \}.

This set consists of all p-adic numbers whose p-adic valuation is non-negative, making \mathbb{Z}_p a subring of \mathbb{Q}_p.^[1]^[16] Equivalently, every element of \mathbb{Z}_p admits a unique representation as a formal power series with coefficients in the digit set \{0, 1, \dots, p-1\} and non-negative powers of p:

\mathbb{Z}_p = \left\{ \sum_{n=0}^\infty a_n p^n : 0 \leq a_n < p \right\}.

This series converges in the p-adic topology, providing a concrete way to visualize p-adic integers as "infinite expansions to the left" in base p.^[1]^[13] Another standard construction of \mathbb{Z}_p is as the inverse limit of the rings of integers modulo powers of p:

\mathbb{Z}_p = \lim_{\leftarrow k} \mathbb{Z}/p^k \mathbb{Z},

where the inverse system is given by the natural projection maps \mathbb{Z}/p^k \mathbb{Z} \to \mathbb{Z}/p^{k-1} \mathbb{Z}. Elements of \mathbb{Z}_p are thus equivalence classes of sequences (a_k)_{k \geq 1} with a_k \in \mathbb{Z}/p^k \mathbb{Z} such that a_k \equiv a_{k-1} \pmod{p^{k-1}} for each k. These two constructions—the power series and inverse limit—are isomorphic, bridging analytic and algebraic perspectives.^[13]^[16] The ring of ordinary integers \mathbb{Z} embeds naturally and densely into \mathbb{Z}_p, either by mapping each integer m to the constant sequence (\overline{m}, \overline{m}, \dots) in the inverse limit (where \overline{m} denotes the class modulo p^k) or to the finite power series \sum_{n=0}^N a_n p^n padded with infinite zeros. The multiplicative units in \mathbb{Z}_p are the elements with p-adic absolute value exactly 1:

\mathbb{Z}_p^\times = \{ x \in \mathbb{Z}_p : |x|_p = 1 \},

which form a group under multiplication. A distinguished subgroup consists of the principal units $1 + p \mathbb{Z}_p, comprising elements congruent to 1 modulo p.^[1]^[13]

Basic properties

The p-adic integers \mathbb{Z}_p are compact in the p-adic topology, a property arising from their construction as the inverse limit of the rings \mathbb{Z}/p^n\mathbb{Z}.^[16] This compactness, combined with the totally disconnected nature of the topology, renders \mathbb{Z}_p homeomorphic to the Cantor set.^[21] Algebraically, \mathbb{Z}_p serves as the ring of integers of the field of p-adic numbers \mathbb{Q}_p, defined as the set of elements with p-adic valuation at least zero.^[16] It is integrally closed in \mathbb{Q}_p, meaning every element of \mathbb{Q}_p that satisfies a monic polynomial with coefficients in \mathbb{Z}_p already belongs to \mathbb{Z}_p.^[16] As an integrally closed domain, \mathbb{Z}_p admits unique factorization of its nonzero nonunit elements into prime elements, up to units in \mathbb{Z}_p.^[22] The ideal p\mathbb{Z}_p forms the unique maximal ideal of \mathbb{Z}_p, generated by the prime p.^[16] This structure makes \mathbb{Z}_p a discrete valuation ring (DVR), characterized by its principal ideals p^n \mathbb{Z}_p for n \geq 0 and the fact that every nonzero ideal is of this form.^[22] The residue field of \mathbb{Z}_p modulo its maximal ideal is isomorphic to the finite field \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}.^[16]

Topological Properties

p-adic metric and valuation

The p-adic valuation on the field of p-adic numbers \mathbb{Q}_p is a function v_p: \mathbb{Q}_p \to \mathbb{Z} \cup \{\infty\} that assigns to each non-zero element an integer measuring its "divisibility" by powers of the prime p, with v_p(0) = \infty.^[2] It satisfies the properties v_p(xy) = v_p(x) + v_p(y) for all x, y \in \mathbb{Q}_p and v_p(x + y) \geq \min(v_p(x), v_p(y)) for all x, y \in \mathbb{Q}_p, with equality in the second property holding when v_p(x) \neq v_p(y).^[18] This valuation extends the p-adic valuation on the rational numbers \mathbb{Q}, defined initially for non-zero integers n as the highest power of p dividing n, and then for rationals a/b as v_p(a) - v_p(b).^[23] The p-adic metric is induced by the normalized absolute value |\cdot|_p: \mathbb{Q}_p \to [0, \infty) given by |x|_p = p^{-v_p(x)} for x \neq 0 and |0|_p = 0, which satisfies |xy|_p = |x|_p |y|_p and the ultrametric inequality |x + y|_p \leq \max(|x|_p, |y|_p) for all

x, y \in \mathbb{Q}_p&#36;, with equality if

|x|_p \neq |y|_p. This inequality implies that \mathbb{Q}_pis a non-Archimedean valued field, meaning that for every integern \geq 1, |n|p \leq 1, in contrast to the Archimedean real absolute value where |n|\infty = n \to \inftyasn \to \infty$.^[2] Open balls in this metric are defined as B(x, r) = \{ y \in \mathbb{Q}_p : |y - x|_p < r \} for x \in \mathbb{Q}_p and r > 0, and they coincide with their closures due to the ultrametric property, making every ball both open and closed (clopen).^[18] Spheres, or sets where |y - x|_p = r, similarly exhibit strong clustering properties under the ultrametric, as any two points in such a set can be connected by a chain where distances do not exceed the maximum.^[23]

Topology and completeness

The p-adic topology on the field of p-adic numbers \mathbb{Q}_p is generated by the open balls defined via the p-adic metric, which satisfies the ultrametric inequality. This metric induces a Hausdorff topology on \mathbb{Q}_p, as distinct points can be separated by disjoint open sets.^[24] The basis for this topology consists of clopen balls, meaning every open ball is both open and closed due to the strong triangle inequality of the metric.^[24] Consequently, \mathbb{Q}_p is totally disconnected: the only connected subsets are singletons, as any larger set can be partitioned into disjoint nonempty clopen subsets.^[24]^[25] By construction, \mathbb{Q}_p is the metric completion of the rational numbers \mathbb{Q} with respect to the p-adic metric, ensuring that every Cauchy sequence in \mathbb{Q}_p converges to an element within \mathbb{Q}_p.^[24]^[26] This completeness distinguishes \mathbb{Q}_p from \mathbb{Q}, which is incomplete under the same metric. The subspace of p-adic integers \mathbb{Z}_p = \{ x \in \mathbb{Q}_p : |x|_p \leq 1 \} is compact in this topology, as it arises as the inverse limit of the finite rings \mathbb{Z}/p^n \mathbb{Z} equipped with the discrete topology, yielding a compact, totally disconnected space.^[26]^[25] Since \mathbb{Z}_p is a compact open neighborhood of the identity, \mathbb{Q}_p itself is locally compact.^[26] As a locally compact topological group under addition, \mathbb{Q}_p admits a unique (up to positive scalar multiple) nonzero left-invariant Haar measure \mu, which can be normalized so that \mu(\mathbb{Z}_p) = 1.^[27] Topologically, \mathbb{Q}_p is homeomorphic to the countably infinite product \prod_{n \in \mathbb{Z}} \{0, 1, \dots, p-1\} endowed with the product topology, where each factor carries the discrete topology; this reflects the bidirectional infinite p-adic expansions of elements in \mathbb{Q}_p.^[26]^[24]

Algebraic Properties

Cardinality and field structure

The field of p-adic numbers, \mathbb{Q}_p, is equipped with addition and multiplication operations that extend those from \mathbb{Q}, satisfying the field axioms: associativity, commutativity, distributivity, existence of additive and multiplicative identities (0 and 1, respectively), and additive inverses (negatives). For every non-zero element x \in \mathbb{Q}_p, a multiplicative inverse x^{-1} exists, ensuring the structure forms a field; this inverse can be constructed explicitly using geometric series expansions when x is a unit in the p-adic integers \mathbb{Z}_p, as non-zero elements are of the form p^k u with k \in \mathbb{Z} and u \in \mathbb{Z}_p^\times, yielding x^{-1} = p^{-k} u^{-1}.^[28] Like its subfield \mathbb{Q}, \mathbb{Q}_p has characteristic 0, meaning no positive integer n satisfies n \cdot 1 = [0](/page/0).^[29] The cardinality of \mathbb{Q}_p is |\mathbb{Q}_p| = 2^{\aleph_0} = \mathfrak{c}, the cardinality of the continuum, matching that of the real numbers \mathbb{R}. This follows from the fact that the p-adic integers \mathbb{Z}_p have cardinality \mathfrak{c}, as \mathbb{Z}_p is uncountable and can be shown to have the same size as the Cantor set via its representation as formal power series in p with coefficients in \{0, 1, \dots, p-1\}, and \mathbb{Q}_p is a countable union \mathbb{Q}_p = \bigcup_{n \in \mathbb{Z}} p^n \mathbb{Z}_p, preserving the cardinality under countable unions of sets of size \mathfrak{c}.^[24] Consequently, \mathbb{Q}_p is uncountable, distinguishing it sharply from the countable dense subfield \mathbb{Q}. While \mathbb{Q}_p shares the characteristic 0 property with \mathbb{Q}, its completeness with respect to the p-adic metric provides a topological foundation that enables the convergence of series defining field operations and inverses, unlike the incomplete \mathbb{Q}. Additionally, \mathbb{Z}_p is a compact topological ring of cardinality \mathfrak{c}, reinforcing the infinite extension nature of \mathbb{Q}_p over \mathbb{Q}.^[24]

Algebraic closure and extensions

Finite extensions of the field \mathbb{Q}_p of p-adic numbers are generated by adjoining a root \alpha of an irreducible polynomial f(x) \in \mathbb{Q}_p, resulting in a field K = \mathbb{Q}_p(\alpha) with degree [K : \mathbb{Q}_p] = n = \deg f.^[30] For such an extension K/\mathbb{Q}_p, the degree n factors as n = e f, where e is the ramification index and f is the residue degree.^[30] The ramification index e is defined as the index [v(K^\times) : v(\mathbb{Q}_p^\times)], where v denotes the normalized additive valuation on \mathbb{Q}_p extended to K, measuring the ramification of the maximal ideal in the ring of integers of K.^[30] The residue degree f is the degree [\kappa_K : \mathbb{F}_p] of the residue field \kappa_K of K over the prime field \mathbb{F}_p.^[30] Unramified extensions correspond to extensions of the residue field: an unramified extension of degree n has residue field \mathbb{F}_{p^n} and ramification index e = 1.^[31] The maximal unramified extension \mathbb{Q}_p^\mathrm{nr} of \mathbb{Q}_p is the fixed field of the inertia subgroup in the absolute Galois group and has residue field equal to the algebraic closure \overline{\mathbb{F}}_p of \mathbb{F}_p, which is an infinite, separable extension obtained as the union \bigcup_{n=1}^\infty \mathbb{F}_{p^n}.^[31] This maximal unramified extension is Galois over \mathbb{Q}_p with Galois group isomorphic to \hat{\mathbb{Z}}, the profinite completion of \mathbb{Z}.^[32] The algebraic closure \overline{\mathbb{Q}}_p of \mathbb{Q}_p is the union of all finite extensions of \mathbb{Q}_p and thus has cardinality |\overline{\mathbb{Q}}_p| = \mathfrak{c}^{\aleph_0} = 2^{\aleph_0}, the same as that of \mathbb{Q}_p since adjoining algebraic elements does not increase the cardinality of an infinite field.^[33] However, \overline{\mathbb{Q}}_p equipped with the extension of the p-adic valuation is not complete, as it fails to be a Baire space.^[34] Its completion \mathbb{C}_p is both complete and algebraically closed, serving as the p-adic analogue of the complex numbers.^[34] While \overline{\mathbb{Q}}_p consists solely of algebraic elements over \mathbb{Q}_p, the full field \mathbb{C}_p includes transcendental extensions, which exhibit a transcendence degree of $2^{\aleph_0} over \mathbb{Q}_p, analogous to the real numbers [\mathbb{R}](/page/The_Real) over \mathbb{Q} but with a non-Archimedean uniformity in the valuation topology.^[35]

Arithmetic Operations

Addition and multiplication

p-adic numbers are represented as formal Laurent series \sum_{n = k}^\infty a_n p^n, where k \in \mathbb{Z}, a_n \in \{0, 1, \dots, p-1\}, and only finitely many negative powers are nonzero.^[13] Addition of two p-adic numbers \alpha = \sum_{n = k}^\infty a_n p^n and \beta = \sum_{n = k}^\infty b_n p^n (aligning by the lowest power if necessary) proceeds by adding corresponding coefficients digit-wise, starting from the lowest power, and propagating carries to higher powers. For each power n, the initial sum is s_n = a_n + b_n + c_n, where c_n is the carry from the previous power (with c_k = 0); the coefficient is then d_n = s_n \mod p, and the carry to the next power is c_{n+1} = \lfloor s_n / p \rfloor. This process ensures the result is again a valid p-adic expansion, as carries propagate only finitely far to the right in practice for the completion, but formally defines the operation componentwise in the inverse limit construction.^[36]^[13] For example, in the 2-adic numbers, adding 1 (expansion \dots 0001_2) and 1 yields \dots 0010_2 = 2, with a carry propagating from the units place.^[1] Multiplication is defined analogously to polynomial multiplication on the series expansions, followed by normalization via carries. For \alpha = \sum_{n = k}^\infty a_n p^n and \beta = \sum_{n = m}^\infty b_n p^n, the product is \gamma = \sum_{n = k+m}^\infty c_n p^n, where the initial coefficients are c_n = \sum_{i + j = n} a_i b_j; since only finitely many pairs (i, j) contribute to each fixed n (due to the finite support in negative powers), each c_n is a finite sum. These coefficients are then adjusted by carrying: for each n starting from the lowest, set d_n = c_n \mod p and add \lfloor c_n / p \rfloor to c_{n+1}, repeating until all coefficients are in \{0, 1, \dots, p-1\}. This yields a unique p-adic expansion, and the operation is well-defined because the finite contributions per power ensure convergence in the p-adic topology.^[36]^[1] For instance, in the 3-adic numbers, multiplying 2 (expansion \dots 0002_3) by 2 gives initial coefficients leading to \dots 0011_3 = 4 after carrying the sum 4 mod 3 = 1 with carry 1 to the next power.^[37] These operations make the p-adic numbers \mathbb{Q}_p into a field, with no zero divisors, as the ring of p-adic integers \mathbb{Z}_p is an integral domain and every nonzero element has a multiplicative inverse.^[13]^[38]

Hensel's lemma and lifting

Hensel's lemma provides a method for lifting solutions of polynomial congruences modulo a prime p to solutions in the p-adic integers \mathbb{Z}_p. In its basic form, the lemma states that if f(X) \in \mathbb{Z}[X] is a polynomial with integer coefficients, a \in \mathbb{Z} satisfies f(a) \equiv 0 \pmod{p} and f'(a) \not\equiv 0 \pmod{p}, then there exists a unique b \in \mathbb{Z}_p such that f(b) = 0 and b \equiv a \pmod{p}.^[39] This result, originally formulated by Kurt Hensel in the context of p-adic analysis, ensures that simple roots modulo p extend uniquely to p-adic roots.^[40] A more general version of Hensel's lemma uses the p-adic valuation v_p. For f(X) \in \mathbb{Z}_p[X] and a \in \mathbb{Z}_p, if v_p(f(a)) > 2 v_p(f'(a)), then there exists a unique \alpha \in \mathbb{Z}_p such that f(\alpha) = 0 and v_p(\alpha - a) > v_p(f'(a)). If v_p(f(a)) = 2 v_p(f'(a)), lifting is possible but may not be unique, depending on higher-order conditions such as the valuation of the second derivative or further terms in the Taylor expansion. These conditions generalize the basic case, where v_p(f'(a)) = 0 and v_p(f(a)) \geq 1, allowing solutions even when the derivative is divisible by powers of p.^[39] The proof relies on a p-adic analogue of Newton-Raphson iteration. Starting with an initial approximation a_0 = a, define the sequence

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

in \mathbb{Q}_p. Under the lemma's hypotheses, this sequence converges p-adically to a root \alpha \in \mathbb{Z}_p with f(\alpha) = 0, and the error satisfies v_p(a_{n+1} - \alpha) > 2 v_p(a_n - \alpha), ensuring quadratic convergence in the p-adic metric. The key is that division by f'(a_n) remains well-defined and invertible in \mathbb{Z}_p due to the non-vanishing derivative condition, propagated through the iteration.^[39] For example, consider finding a square root of 7 in the 3-adic integers using f(X) = X^2 - 7. Modulo 3, $7 \equiv 1 \pmod{3} and a = 1 satisfies f(1) = -6 \equiv 0 \pmod{3}, while f'(1) = 2 \not\equiv 0 \pmod{3}. Hensel's lemma guarantees a unique lift to b \in \mathbb{Z}_3 with b^2 = 7 and b \equiv 1 \pmod{3}. The Newton iteration yields approximations starting with b_0 = 1, b_1 = 4 \equiv 1 \pmod{3}, converging to b = 1 + 3 + 3^2 + 2 \cdot 3^4 + \cdots.^[39]

Advanced Structures

Multiplicative group

The multiplicative group \mathbb{Q}_p^\times of the field of p-adic numbers \mathbb{Q}_p comprises all nonzero elements equipped with the operation of multiplication. Every element x \in \mathbb{Q}_p^\times admits a unique decomposition x = p^v \cdot u, where v = v_p(x) \in \mathbb{Z} is the p-adic valuation of x and u \in \mathbb{Z}_p^\times is a p-adic unit (i.e., an element of \mathbb{Z}_p with valuation zero).^[41] This polar decomposition induces a group isomorphism \mathbb{Q}_p^\times \cong \mathbb{Z} \times \mathbb{Z}_p^\times, where the factor \mathbb{Z} arises from the discrete valuation subgroup \{p^k \mid k \in \mathbb{Z}\}. The structure of the unit group \mathbb{Z}_p^\times varies depending on whether p is an odd prime or p=2. For an odd prime p, \mathbb{Z}_p^\times is isomorphic to the direct product of a finite cyclic group of order p-1 (generated by a primitive (p-1)-th root of unity in \mathbb{Z}_p) and the additive group \mathbb{Z}_p (via the isomorphism between $1 + p\mathbb{Z}_p and \mathbb{Z}_p induced by the p-adic logarithm or exponential). More precisely, \mathbb{Z}_p^\times \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p. For the case p=2, the group \mathbb{Z}_2^\times is isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}_2, where the \mathbb{Z}/2\mathbb{Z} factor is generated by -1 and the \mathbb{Z}_2 factor corresponds to $1 + 4\mathbb{Z}_2 under the exponential map. Topologically, the p-adic topology on \mathbb{Q}_p^\times reflects the decomposition: the subgroup isomorphic to \mathbb{Z} (powers of p) carries the discrete topology, as neighborhoods around distinct powers are separated by the ultrametric property. In contrast, \mathbb{Z}_p^\times is compact, being an open subgroup of the compact additive group \mathbb{Z}_p. Thus, \mathbb{Q}_p^\times is a disjoint union of countably many compact cosets of \mathbb{Z}_p^\times. As a locally compact abelian group, \mathbb{Q}_p^\times possesses a unique (up to positive scalar multiple) translation-invariant Haar measure \mu, which is used in p-adic integration theory. This measure is typically normalized such that \mu(\mathbb{Z}_p^\times) = 1, ensuring that integrals over \mathbb{Q}_p^\times align with those over the compact unit subgroup and the discrete valuation factor. Under the isomorphism \mathbb{Q}_p^\times \cong \mathbb{Z} \times \mathbb{Z}_p^\times, the Haar measure decomposes as the product of counting measure on \mathbb{Z} and the normalized measure on \mathbb{Z}_p^\times.

Local-global principle

The local-global principle, also known as the Hasse principle, posits that a Diophantine equation over the rational numbers \mathbb{Q} has a nontrivial solution in \mathbb{Q} if and only if it has solutions in the real numbers \mathbb{R} and in the p-adic numbers \mathbb{Q}_p for every prime p. This principle leverages the completions of \mathbb{Q} at all places (finite primes and the infinite place corresponding to \mathbb{R}) to bridge local solvability with global existence. In the context of p-adic numbers, local solvability requires the equation to admit solutions in each \mathbb{Q}_p, reflecting the ultrametric topology and valuation structure that allow Henselian lifting in many cases.^[42] A prominent success of the Hasse principle occurs for quadratic forms. The Hasse-Minkowski theorem states that a quadratic form over \mathbb{Q} represents zero nontrivially if and only if it does so over \mathbb{R} and over every \mathbb{Q}_p. This equivalence relies on the classification of quadratic forms over local fields, where isotropy (nontrivial zero representation) is determined by dimension, discriminant, and Hasse invariant, enabling a global aggregation via local invariants. The theorem, proved by Hasse in 1921 for \mathbb{Q} and extended by Minkowski, underscores the role of p-adic solvability as a complete set of local conditions for this class of equations.^[43] However, the Hasse principle fails for more general equations, such as certain cubic forms, where local solutions exist everywhere but no global rational solution does. A classic counterexample is Selmer's curve defined by the equation $3x^3 + 4y^3 + 5z^3 = 0, which has nontrivial solutions in \mathbb{R} and in \mathbb{Q}_p for all primes p, yet no nontrivial rational points. This failure highlights that p-adic local conditions, while necessary, are insufficient for global solvability in higher degrees. To formalize such local-global interactions, the adele ring \mathbb{A}_\mathbb{Q} is introduced as the restricted direct product \prod_v' \mathbb{Q}_v over all places v (primes p and \infty), where the restriction ensures components at almost all finite places lie in the p-adic integers \mathbb{Z}_p. The field \mathbb{Q} embeds diagonally into \mathbb{A}_\mathbb{Q}, and rational points correspond to adele points fixed under the Galois action or projecting to the diagonal image, providing a framework to study when local solutions aggregate globally.^[44]^[45] Beyond basic local-global compatibility, failures like Selmer's example are often explained by more refined obstructions, such as the Brauer-Manin obstruction. This obstruction arises from the Brauer group of the variety, measuring cohomological descent data over the adeles; specifically, the pairing of adele points with Brauer classes can detect incompatibilities not visible at the level of individual p-adic solutions. Introduced by Manin in 1970 for cubic surfaces and generalized, it provides a necessary condition for the existence of rational points: if the Brauer-Manin set is empty, the Hasse principle fails, as seen in Selmer's curve where transcendental Brauer elements obstruct global points despite local ubiquity. While not always sufficient, this p-adically informed tool captures many counterexamples and motivates further study of adelic cohomology in number theory.^[46]

Applications

In number theory

p-adic numbers play a crucial role in number theory through the construction of p-adic zeta functions, which enable the interpolation of special values of the classical Riemann zeta function at negative integers. Introduced by Kubota and Leopoldt, the p-adic zeta function \zeta_p(s) is a continuous function on the p-adic integers \mathbb{Z}_p that interpolates the values \zeta(1-k) = -\frac{B_k}{k} for positive integers k \geq 1 not divisible by p-1, where B_k are the Bernoulli numbers. This interpolation relies on Kummer's congruences for Bernoulli numbers, which ensure the values are p-adically continuous, allowing the extension from rational points to the entire p-adic domain. Such p-adic interpolation provides a framework for generalizing classical results, including analogs of Euler's theorem on sums of powers, where p-adic limits replace real summation to express power sums in terms of interpolated Bernoulli numbers.^[47] In class field theory, p-adic numbers underpin local reciprocity laws and the explicit construction of abelian extensions of \mathbb{Q}_p. Lubin and Tate developed a theory of formal groups over local fields, associating to each uniformizer \pi a formal Lubin-Tate group whose torsion points generate explicit totally ramified abelian extensions of \mathbb{Q}_p. This construction realizes the local Artin reciprocity map, identifying the multiplicative group \mathbb{Q}_p^\times with the Galois group of the maximal abelian extension, thus providing a geometric realization of local class field theory. The Lubin-Tate approach not only proves the existence of these extensions but also computes their conductors and ramification, offering tools for studying global class field theory via local completions. Iwasawa theory extends these ideas to infinite towers of cyclotomic fields, using p-adic L-functions to relate class groups and units. Iwasawa constructed p-adic L-functions for the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q}, showing that the p-part of the class group in these fields forms a finitely generated \mathbb{Z}_p[[ \Gamma ]]-module, where \Gamma is the Galois group. The characteristic ideal of this module is generated by the p-adic L-function L_p(s, \chi), which interpolates twisted zeta values and satisfies the main conjecture linking analytic and algebraic structures.^[48] This framework has profound implications for understanding the distribution of primes and the Birch and Swinnerton-Dyer conjecture in elliptic curves. In Diophantine approximation, p-adic continued fractions provide a tool for studying how well algebraic numbers can be approximated by rationals in the p-adic metric. Schneider introduced p-adic continued fraction expansions for elements of \mathbb{Q}_p, analogous to real continued fractions, where the algorithm uses the p-adic valuation to generate partial quotients. These expansions characterize quadratic irrationals and yield effective measures of approximation, such as bounds on | \alpha - r/s |_p < c / |s|_p^\tau for algebraic \alpha, aiding in transcendence proofs and effective versions of Roth's theorem in the p-adic setting.

In analysis and geometry

p-adic analysis provides a framework for studying analytic functions over p-adic fields, analogous to complex analysis but adapted to the non-Archimedean valuation. In this setting, power series converge on disks defined by the p-adic topology, particularly on the closed unit disk \mathbb{Z}_p, where the valuation is bounded by 1. A power series \sum a_n x^n with coefficients in \mathbb{Q}_p converges for x \in \mathbb{Z}_p if the coefficients satisfy |a_n| \to 0 as n \to \infty, ensuring uniform convergence on compact subsets like \mathbb{Z}_p. This convergence property allows for the definition of analytic functions on p-adic spaces, differing from real analysis where convergence radii are determined by Archimedean norms. The Tate algebra \mathbb{Q}_p \langle T_1, \dots, T_n \rangle consists of power series in several variables that converge on the unit polydisk \mathbb{Z}_p^n, equipped with the Gauss norm \|f\| = \sup |a_\alpha| over multi-indices \alpha. These algebras are Banach spaces over \mathbb{Q}_p and form the building blocks of rigid analytic spaces, introduced by John Tate as a rigid version of analytic spaces to avoid pathologies in non-Archimedean geometry. Rigid analytic spaces are locally affine schemes over Tate algebras, glued along admissible open sets, enabling the study of analytic continuation and maximum modulus principles in the p-adic context. For instance, the rigid analytic affine line \mathbb{A}^1_{\mathrm{rig}} over \mathbb{Q}_p covers the p-adic plane minus certain points, capturing geometric properties like the absence of non-constant bounded holomorphic functions on the whole space.^[49] Integration in p-adic analysis relies on the Haar measure on \mathbb{Q}_p, which is a translation-invariant, locally finite measure on the additive group (\mathbb{Q}_p, +). Normalized such that the measure of \mathbb{Z}_p is 1, the Haar measure \mu satisfies \mu(p \mathbb{Z}_p) = p^{-1} and extends multiplicatively: for a \in \mathbb{Q}_p^\times, \mu(a U) = |a|_p \mu(U) for measurable sets U. This measure enables the integration of continuous functions f: \mathbb{Q}_p \to \mathbb{C}, defined as limits of integrals over compact sets, and supports Fubini's theorem for products. Additive characters play a crucial role in Fourier analysis over \mathbb{Q}_p; the standard nontrivial character is \psi(x) = \exp(2 \pi i \{x\}), where \{x\} denotes the fractional part of x with respect to the p-adic expansion, trivial on \mathbb{Z}_p but not on \mathbb{Q}_p. The Fourier transform \hat{f}(\xi) = \int_{\mathbb{Q}_p} f(x) \psi(-x \xi) \, d\mu(x) inverts functions in Schwartz-Bruhat spaces, facilitating harmonic analysis and local zeta functions.^[50]^[51] In arithmetic geometry, p-adic Hodge theory bridges étale, de Rham, and crystalline cohomologies, providing comparison isomorphisms that relate the p-adic topology to differential structures. De Rham cohomology H^*_{\mathrm{dR}}(X/\mathbb{Q}_p) of a smooth proper variety X over \mathbb{Q}_p carries a Hodge filtration from the de Rham complex, while crystalline cohomology H^*_{\mathrm{crys}}(X/W(k)) \otimes \mathbb{Q}_p arises from lifts to characteristic zero via divided power envelopes, capturing Frobenius actions in positive characteristic reductions. Fontaine's theory establishes that for representations of \mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p), weakly admissible filtered \phi-modules correspond to de Rham representations, with crystalline ones forming a subcategory where the Hodge filtration lies within the image of Frobenius. These comparisons, such as the crystalline comparison theorem, equate H^*_{\mathrm{ét}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Q}_p) \otimes B_{\mathrm{crys}} \cong H^*_{\mathrm{crys}}(X/W) \otimes B_{\mathrm{crys}}, enabling the study of period rings like B_{\mathrm{dR}} and B_{\mathrm{crys}} in mixed-characteristic settings. This framework is essential for proving properties like the Fontaine-Messing conjecture on weakly admissible modules.^[52]^[53] Teichmüller lifts provide canonical embeddings from the algebraic closure \overline{\mathbb{F}}_p of the finite field \mathbb{F}_p into the p-adic integers \mathbb{Z}_p, or more generally into the ring of Witt vectors W(\overline{\mathbb{F}}_p), preserving multiplicative structure in characteristic p. For \alpha \in \overline{\mathbb{F}}_p, the Teichmüller lift [\alpha] \in W(\overline{\mathbb{F}}_p) is the unique root of the polynomial X^p - X - \alpha = 0 in the Witt vectors, satisfying [\alpha]^p = [\alpha] and lifting the residue field. These lifts extend to characters via the Teichmüller character \omega: \mathbb{F}_p^\times \to \mathbb{Z}_p^\times, which is the unique unramified character of order p-1. In arithmetic geometry, Teichmüller lifts facilitate the construction of canonical lifts of varieties from characteristic p to mixed characteristic, such as in the study of abelian varieties with potentially good reduction, and appear in the decomposition of units \mathbb{Z}_p^\times \cong \mu_{p-1} \times (1 + p \mathbb{Z}_p).^[54]^[55]

In physics and other fields

p-adic quantum mechanics extends the formalism of standard quantum mechanics by replacing real numbers with p-adic numbers for canonical variables, allowing the study of free particles and harmonic oscillators where time can be p-adic while coordinates and momenta are either p-adic or real.^[56] This framework constructs quantum mechanics using complex-valued wave functions in the Hilbert space L^2(\mathbb{Q}_p), employing the Weyl representation for operators.^[56] The evolution operator for the harmonic oscillator is explicitly derived, solving a p-adic analog of the Schrödinger equation, with generalized vacuum states defined for primes p = 4\ell + 1.^[56] In string theory, p-adic and adelic formulations unify real and p-adic descriptions by factoring Veneziano and Virasoro-Shapiro amplitudes into products over non-Archimedean contributions, providing a number-theoretic perspective on string scattering.91357-8) The adelic string approach treats the worldsheet and spacetime on equal footing across all completions of the rationals, enabling simultaneous analysis of ordinary and p-adic strings.91357-8) More recently, p-adic AdS/CFT correspondence posits a duality between a conformal field theory on the boundary \mathbb{Q}_p (or its extension) and bulk gravity on the p-adic hyperbolic space, modeled via the Bruhat-Tits tree, offering insights into holographic principles in non-Archimedean geometries. Beyond physics, p-adic numbers inform computational models in computer science, such as relaxed algorithms for exact p-adic arithmetic in computer algebra systems, which optimize precision control and storage for numerical simulations and cryptographic applications. In biology, p-adic models describe the genetic code and DNA sequences by encoding nucleotides and codons in \mathbb{Q}_p, revealing hierarchical structures and degeneration patterns as p-adic phenomena, with diffusion-like processes modeling evolutionary dynamics in genomic spaces. These applications remain non-standard, as physical observables are inherently real-valued and require embeddings from \mathbb{Q}_p to \mathbb{R} or \mathbb{C} for empirical connection, yet they provide valuable hints for quantum gravity by suggesting discreteness at the Planck scale through adelic unification.

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[PDF] p-adic numbers, quadratic forms, and the hasse-minkowski theorem
Nov 10, 2021 · Motivated by the analogies between the number fields and the function fields, Kurt. Hensel introduced a whole new number field—the p-adic ...
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[PDF] Elliptic curves - The Library at SLMath
May 2, 2000 · Hasse Principle can fail for plane curves of degree 3. Here is a counterexample due to Selmer [1951; 1954]: the curve 3X. 3 C 4Y 3 C 5Z3 D 0 in ...
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[PDF] Adeles and Ideles and their applications - UChicago Math
Aug 18, 2010 · is homeomorphic to f−1(1) and since f is trivially continuous the ... Then E(P) is the direct product of E and a group isomorphic to Zs ...
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[PDF] brauer–manin obstructions requiring arbitrarily many brauer classes
All but a handful of such examples in the literature are explained by the Brauer–Manin obstruction, first introduced by Manin in. 1970 [Man71]. Manin observed ...
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p-adic zeta functions and Bernoulli numbers
One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-functi.
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https://press.princeton.edu/books/paperback/9780691081120/lectures-on-p-adic-l-functions
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[PDF] 1. Affinoid algebras and Tate's p-adic analytic spaces : a brief survey
The convergence property of such a power series exactly ensures that it can be evaluated on Zm p . The elements of the Tate algebra are called a rigid.
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[PDF] Course Notes for Math 574: Adeles, Automorphic Forms, and ...
Jan 23, 2002 · This gives a tree-like structure to Qp. 2.2 Haar Measure and Integration. There is an additive measure dx, normalized to give Zp measure 1, and ...
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[PDF] Fourier_Analysis_on_Number_Fi...
Then by definition of the Fourier transform and Haar measure, it is immediate ... standard additive character If/p on Qp defined by the following composition:.
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[PDF] arXiv:2005.07919v1 [math.NT] 16 May 2020
May 16, 2020 · p-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review p-adic Hodge Theory for ...
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[PDF] p-adic Hodge theory for rigid-analytic varieties
Nov 3, 2012 · Introduction. This paper starts to investigate to what extent p-adic comparison theorems stay true for rigid-analytic varieties.
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[PDF] Several approaches to non-archimedean geometry - Mathematics
Tate algebras. In this first lecture, we discuss the commutative algebra that forms the foundation for the local theory of rigid-analytic spaces, much as.Missing: Z_p | Show results with:Z_p
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[PDF] Neal Koblitz - p-adic Numbers, p-adic Analysis, and Zeta-Functions
Koblitz, Neal, 1948-. P-adic numbers, p-adic analysis and zeta-functions. (Graduate texts in mathematics; 58). Bibliography: p. Includes index. l. p-adic ...
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https://doi.org/10.1007/BF01218590