Glossary

11.1 p-adic numbers and fields

4 min read•july 30, 2024

P-adic numbers are a fascinating twist on our usual number system. They're built using a different way of measuring "closeness" based on divisibility by a prime number. This leads to some wild results, like 9 and 1 being "close" in the 2-adic world!

These numbers form their own complete field, Q_p, with unique properties. They're crucial in number theory, helping solve tricky equations and offering new perspectives on old problems. Plus, they connect to broader ideas about and completions.

P-adic Absolute Value

Definition and Basic Properties

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  • P-adic absolute value operates as a non-Archimedean valuation on the rational number field for a prime number p
  • For non-zero rational x, p-adic absolute value |x|_p equals p−ordp(x)p^{-ord_p(x)}, with ordp(x)ord_p(x) representing the highest p power dividing x
  • Satisfies ∣x+y∣p≤max(∣x∣p,∣y∣p)|x + y|_p \leq max(|x|_p, |y|_p) for all x and y
  • Demonstrates multiplicativity ∣xy∣p=∣x∣p∣y∣p|xy|_p = |x|_p|y|_p for all x and y
  • Assigns |0|_p = 0 and |1|_p = 1 for any prime p

Metric and Topology Induced by P-adic Absolute Value

  • Induces a metric on rational numbers, defining distance function d(x,y)=∣x−y∣pd(x,y) = |x - y|_p
  • Establishes unique "closeness" notion compared to standard absolute value
  • Numbers with high p powers in denominators considered "small" (1/p^n for large n)
  • Creates distinct topology on rational numbers compared to usual Euclidean topology
  • Leads to counterintuitive results (9 and 1 considered "close" in 2-adic absolute value)

Applications and Significance

  • Plays crucial role in number theory and algebraic geometry
  • Used in solving Diophantine equations (equations with integer coefficients)
  • Provides tool for studying local properties of algebraic varieties
  • Facilitates p-adic analysis, analogous to real and complex analysis
  • Enables new approaches to classical problems (Fermat's Last Theorem)

P-adic Number Field

Construction and Representation

  • Field of p-adic numbers Q_p constructed as rational number completion with respect to p-adic absolute value
  • Each p-adic number uniquely represented as infinite series ∑n=−k∞anpn\sum_{n=-k}^{\infty} a_n p^n, where an∈{0,1,...,p−1}a_n \in \{0, 1, ..., p-1\} and k is an integer
  • Z_p form subring of Q_p, consisting of p-adic numbers with non-negative
  • Z_p represented by series ∑n=0∞anpn\sum_{n=0}^{\infty} a_n p^n with an∈{0,1,...,p−1}a_n \in \{0, 1, ..., p-1\}

Algebraic Structure and Properties

  • Q_p functions as locally compact topological field (field and topological space with compatibility conditions)
  • Algebraic closure of Q_p, denoted C_p, lacks (unlike complex numbers to real numbers)
  • Q_p contains all p-th roots of unity, crucial for local class field theory
  • Multiplicative group of units in Z_p, Z_p^*, possesses specific structure dependent on p (odd or 2)
  • Q_p exhibits characteristic 0, infinite, and uncountable properties
  • Finite extensions of Q_p studied in local class field theory
  • Unramified extensions of Q_p correspond to finite field extensions
  • P-adic Lie groups generalize real Lie groups in p-adic context
  • Bruhat-Tits buildings provide geometric structure for studying p-adic groups
  • P-adic analysis extends real and complex analysis concepts to p-adic setting

Arithmetic in P-adic Numbers

Basic Operations

  • Addition and subtraction performed digit-by-digit, carrying over when necessary (similar to decimal arithmetic but base p)
  • Multiplication utilizes distributive property and rules for multiplying p powers
  • Division possible for any non-zero p-adic number (every non-zero element has multiplicative inverse)
  • Computation examples:
    • In Q_5: (1 + 25 + 35^2) + (4 + 25) = 0 + 05 + 4*5^2
    • In Q_3: (1 + 23) * (2 + 3) = 2 + 23 + 2*3^2

Advanced Functions and Algorithms

  • P-adic exponential function exp(x) defined for x with ∣x∣p<p−1/(p−1)|x|_p < p^{-1/(p-1)}
  • P-adic logarithm function log(1+x) defined for x with ∣x∣p<1|x|_p < 1
  • Both exp(x) and log(1+x) satisfy properties analogous to real counterparts within domains
  • Square root and higher root computation follows specific algorithms exploiting p-adic expansion
  • provides tool for solving polynomial equations in Z_p
  • Newton's method adapted for p-adic context to find roots of polynomials

Computational Techniques and Applications

  • P-adic expansion used for efficient computation of certain number-theoretic functions
  • Teichmüller representatives employed to simplify certain p-adic calculations
  • P-adic methods applied in factoring algorithms (p-adic factoring method)
  • Local-global principle utilizes p-adic computations to solve problems over rational numbers
  • P-adic period integrals computed in certain cases of p-adic cohomology theories

Topology of P-adic Numbers

Topological Properties

  • Q_p topology characterized as totally disconnected (only connected subsets are single points)
  • Q_p forms complete metric space with respect to p-adic metric (every Cauchy sequence converges)
  • P-adic integers Z_p constitute compact subset of Q_p in p-adic topology
  • Open balls in p-adic topology exhibit unique property: every point in open ball serves as its center
  • Q_p lacks ordering compatible with field operations (unlike real numbers)

Visualizations and Analogies

  • Cantor set provides useful analogy for visualizing Z_p topology
  • Tree structure (specifically p-ary tree) often used to represent p-adic numbers
  • Fractal-like nature of p-adic integers observed in certain representations
  • Adele ring combines all p-adic completions with real numbers, providing global perspective
  • Completion process constructing Q_p from Q generalizes to other fields with absolute values
  • Concept of valued fields emerges from this generalization
  • Non-Archimedean analysis developed as study of analysis over non-Archimedean fields
  • Berkovich spaces provide alternative approach to p-adic geometry
  • Model theory of valued fields connects p-adic numbers to logic and set theory

Key Terms to Review (18)

Cauchy Sequences in p-adics: Cauchy sequences in p-adics are sequences of p-adic numbers where, for any given positive tolerance, there exists a point in the sequence beyond which all subsequent terms are closer together than the specified tolerance. This property is crucial in the context of p-adic numbers because it allows for the establishment of convergence within the p-adic field, meaning that these sequences can effectively lead to a limit that also resides in the same field. Understanding Cauchy sequences is essential for grasping how limits are handled in p-adic analysis, as they provide a framework for defining convergence distinctly from classical real number analysis.
Completeness: Completeness refers to the property of a mathematical system in which every bounded sequence has a limit that exists within that system. In the context of p-adic numbers, completeness means that every Cauchy sequence of p-adic numbers converges to a limit that is also a p-adic number. This ensures that p-adic numbers form a complete field, providing a solid foundation for further exploration in algebraic number theory.
Completion of the Rational Numbers: The completion of the rational numbers is the process of extending the rational number system to include limits of Cauchy sequences, resulting in the real numbers. This concept helps us understand how the rational numbers, while dense, have gaps that are filled by irrational numbers, allowing for a complete number system where every Cauchy sequence converges to a limit within the system.
Finite Extension: A finite extension is a field extension in which the larger field has a finite dimension as a vector space over the smaller field. This concept connects various aspects of algebraic structures, showcasing how algebraic numbers and integers can form fields with finite degrees, and how properties such as norms, traces, and discriminants are integral to understanding these extensions.
Hensel's Lemma: Hensel's Lemma is a fundamental result in p-adic analysis that provides a criterion for lifting solutions of polynomial equations from the residue field to the p-adic integers. It connects the concept of p-adic numbers with algebraic equations, allowing us to find roots in a more refined p-adic setting. This lemma is crucial for understanding local properties of algebraic equations and plays a key role in various advanced concepts such as completions, strong approximations, and local-global principles.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has been pivotal in the development of modern mathematics, particularly in understanding the properties of p-adic numbers and fields as well as the foundational aspects of adele rings, which serve as a unifying framework in number theory.
Kurt Hensel: Kurt Hensel was a prominent German mathematician best known for his contributions to number theory, particularly the development of p-adic numbers and fields. His work laid the foundation for a new way of thinking about numbers, allowing mathematicians to study solutions to polynomial equations with respect to different prime bases. This innovative approach has had a lasting impact on various areas of mathematics, including algebraic geometry and local fields.
Local fields: Local fields are a special class of fields that are complete with respect to a discrete valuation and have finite residue fields. These fields are crucial in number theory, particularly when studying properties of p-adic numbers and how they behave under various arithmetic operations. They also serve as foundational elements when looking at discrete valuations and their corresponding valuation rings, as well as in the context of understanding the Artin reciprocity law, which links local and global fields through Galois theory.
Local-global principles: Local-global principles refer to the idea that a mathematical statement can be verified in local settings (like p-adic numbers) and, under certain conditions, this local verification can imply the truth of the statement in a global setting (like rational numbers). This concept is especially significant in number theory, where understanding properties of numbers in local contexts can provide insights into their behavior over the entire field.
P-adic expansions: p-adic expansions are a way of expressing numbers in terms of powers of a prime number p, leading to a unique representation that allows for arithmetic in the p-adic number system. This method not only helps in representing integers and rational numbers but also plays a crucial role in understanding the structure of p-adic numbers, including their convergence properties and how they interact with classical number theory.
P-adic integers: p-adic integers are a set of numbers that extend the concept of integers in a way that is useful for number theory, particularly in relation to prime numbers. They form a ring that is complete with respect to the p-adic topology, which is defined using a prime number p. This structure allows for a unique way to represent numbers, particularly when dealing with congruences and local properties of numbers in number fields.
P-adic l-functions: p-adic l-functions are a type of analytic function that arise in the study of number theory, particularly in connection with p-adic numbers and the properties of Galois representations. These functions extend classical L-functions to the p-adic setting, playing a crucial role in class field theory, where they are used to investigate the connections between algebraic number fields and their ideal class groups. They also help in understanding the distribution of primes and other arithmetic properties in a p-adic context.
P-adic norm: The p-adic norm is a function that assigns a non-negative value to each rational number based on its divisibility by a prime number p. This norm provides a way to measure the 'size' of numbers in the p-adic number system, emphasizing how close a number is to being divisible by p and allowing for a unique topology on the set of p-adic numbers.
P-adic rationals: P-adic rationals are a system of numbers that extend the traditional notion of rational numbers by introducing a new way of measuring distance based on a prime number p. This system allows for the completion of the rational numbers with respect to the p-adic metric, which focuses on divisibility by p rather than the usual absolute value. P-adic numbers play a crucial role in number theory and algebra, enabling mathematicians to study properties and structures that are not easily accessible through conventional means.
P-adic valuation: The p-adic valuation is a function that assigns to each non-zero rational number a non-negative integer, reflecting how many times that number can be divided by a prime number p before it becomes a unit. This concept is essential in understanding the structure of p-adic numbers and their fields, as it provides a way to measure the 'size' of numbers in a different sense than the usual absolute value. It links deeply with discrete valuations and valuation rings, playing a crucial role in algebraic number theory by providing a means to study the properties of integers and rational numbers in relation to prime factors.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Theorem of Ostrowski: The Theorem of Ostrowski states that every non-Archimedean valuation on the field of rational numbers is equivalent to either the p-adic valuation for some prime p or the trivial valuation. This theorem is fundamental in understanding the structure of valuations on fields, especially in the context of p-adic numbers and their applications in number theory.
Ultrametric inequality: The ultrametric inequality is a property of a function that defines a p-adic metric, stating that for any three points in a p-adic space, the distance between any two points is less than or equal to the maximum of the distances from those points to a third point. This unique form of distance measurement leads to distinct behaviors compared to classical metrics and is foundational in understanding the structure of p-adic numbers and fields.
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