11.2 Completions of number fields
4 min read•july 30, 2024
Completions of number fields are like adding missing puzzle pieces to our number system. They help us understand tricky problems by zooming in on specific parts, just like using a magnifying glass to see tiny details.
These completions give us new tools to solve equations and study number properties. They're like secret passageways that connect different areas of math, helping us uncover hidden patterns and relationships between numbers.
Completions of Number Fields
Completion Process and Properties
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- Completions of number fields extend the field by completing it with respect to a prime ideal, analogous to real numbers completing rational numbers
- Process involves taking the limit of Cauchy sequences in the field with respect to a p-adic , where p associates with the prime ideal
- p-adic absolute value measures divisibility by p, not size in the usual sense
- Resulting completions algebraically closed fields contain the original number field as a dense subfield
- Completion process preserves many algebraic properties (characteristic, degree over base field)
- Allows study of local properties of number fields, often used to deduce global properties through local-global principles
- Concept generalizes to other algebraic structures (rings, groups)
Examples and Applications
- Completion of rational numbers Q with respect to prime ideal (2) yields the field of 2-adic numbers Q2
- Completion of Q(√2) with respect to prime ideal (3) results in a 2-dimensional extension of Q3
- Used in studying factorization of polynomials over p-adic fields (x^2 + 1 factors as (x + i)(x - i) in the 5-adic completion of Q, but not in Q5 itself)
- Applied in solving Diophantine equations (x^2 + y^2 = 3 has no solutions in Q2, therefore no rational solutions)
Local Fields vs Completions
Relationship and Equivalence
- Local fields precisely completions of global fields (number fields or function fields) with respect to discrete valuations
- Every with respect to a prime ideal
- Every local field of characteristic 0 isomorphic to completion of some number field
- Residue field of a local field finite, size related to degree of completion over prime subfield
- Local fields equipped with natural topology induced by absolute value used in completion process
- of local field discrete ring, principal ideal domain with unique maximal ideal
- Theory of local fields provides unified framework for studying p-adic numbers, formal power series, certain valued fields
Galois Theory and Applications
- Galois theory of local fields simpler than global fields, useful for studying ramification and local phenomena
- Local fields used in studying decomposition and inertia groups of Galois extensions
- Provide setting for local class field theory, describing abelian extensions of local fields
- Used in analyzing local factors of zeta functions and L-functions (local Euler factors)
Structure of Completions
Degree and Subfield Structure
- Structure of completion depends on prime ideal used in completion process and structure of original number field
- Completions of Q with respect to prime ideals (p) isomorphic to field of p-adic numbers Qp
- For number field K, completion Kp at prime ideal p of Qp, where p rational prime below p
- Degree [Kp : Qp] equal to product of ramification index e and residue degree f of p over p
- Completed field Kp contains unique unramified subextension of Qp of degree f, called maximal unramified subextension
- Multiplicative group of local field decomposes as product of group of units of its ring of integers and cyclic group generated by uniformizer
- Additive group of local field isomorphic to countable direct sum of additive group of its residue field
Examples and Special Cases
- Completion of Q(√5) at prime ideal above 2 2-dimensional extension of Q2
- Unramified quadratic extension of Q2 field Q2(√-3)
- Totally ramified cubic extension of Q3 field Q3(∛3)
- Structure of Qp×: Zp× × ⟨p⟩, where Zp× units in ring of p-adic integers
- Additive group of Q2 isomorphic to countable direct sum of Z/2Z
Applications of Completions in Number Theory
Problem Solving and Theorems
- Used to study factorization of prime ideals in extensions of number fields through
- Local-global principle reduces certain global problems to collection of local problems in completions
- Play crucial role in class field theory, particularly in formulation and proof of Artin reciprocity law
- Essential in understanding ramification and discriminants of number fields and their extensions
- Used in study of zeta functions and L-functions associated with number fields, analyzing behavior at prime ideals
- Norm residue symbol and Hilbert symbol, important tools in algebraic number theory, defined using completions
- Provide natural setting for studying certain Diophantine equations, as solutions often behave more predictably in p-adic setting
Specific Examples and Techniques
- Hensel's lemma used to factor x^2 - 2 over Q7 into (x - √2)(x + √2)
- Local-global principle applied to determine existence of rational points on elliptic curves
- Completions used in computing Hilbert symbols (a,b)p for quadratic reciprocity
- p-adic analysis of Riemann zeta function reveals information about distribution of prime numbers
- Studying ramification in local fields helps understand global ramification in number field extensions
Key Terms to Review (18)
Absolute value: Absolute value is a mathematical concept that represents the non-negative distance of a number from zero on the number line, regardless of its sign. It is denoted by two vertical bars around a number, for example, $$|x|$$, and is crucial in various mathematical contexts, including number fields. Understanding absolute value helps in analyzing properties of numbers and their relationships within the framework of completions.
Arithmetic topology: Arithmetic topology is a branch of mathematics that studies the connections between number theory and topology, particularly in understanding how algebraic properties of number fields relate to topological structures. This concept becomes particularly relevant when considering completions of number fields, where local properties can reveal significant insights into global behaviors. The interactions between arithmetic and topological methods can lead to profound results in both areas.
Completion Map: A completion map is a function that takes a number field and produces its completion with respect to a specific valuation, typically resulting in a field that behaves well under certain algebraic operations. This process allows for the analysis of number fields in a more manageable setting, where issues of convergence and limits can be better understood. Completions help to study the properties of number fields and their extensions by providing a clearer view of local behavior at various places.
Completion of a number field: The completion of a number field is the process of creating a complete metric space from a number field by adding limits to Cauchy sequences, allowing for a richer structure where every Cauchy sequence converges. This concept connects deeply with local fields and p-adic numbers, as it enables the examination of algebraic properties in a more refined manner, accommodating the notion of 'closeness' and enabling various extensions of the field.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Embedding: In the context of algebraic number theory, embedding refers to a homomorphism from one field into another that preserves the structure of the fields involved. Specifically, it involves mapping elements from a subfield or an extension field into a larger field while maintaining operations such as addition and multiplication. This concept is crucial in understanding the relationships between number fields, their completions, and various existence and uniqueness theorems.
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.
Inertia group: An inertia group is a subgroup of the Galois group that describes how primes split in a field extension and identifies the behavior of the ramification of those primes. It plays a crucial role in understanding how primes behave under various extensions, particularly in relation to decomposition and ramification. Inertia groups provide insight into the local behavior of field extensions, connecting to concepts like Frobenius automorphisms and completions of number fields.
Integral Closure: Integral closure refers to the set of all elements in a given field that are integral over a specified ring, particularly focusing on algebraic integers. It connects various concepts like algebraic numbers and integers, providing a way to understand the structure of rings of integers in number fields, ensuring that algebraic properties are preserved within extensions.
Isomorphism: Isomorphism refers to a structural similarity between two algebraic structures that allows for a one-to-one correspondence between their elements while preserving the operations defined on those structures. This concept is essential in various areas of mathematics, as it highlights the inherent equivalence between different algebraic systems, showing how they can behave identically despite potentially differing appearances.
Local Field: A local field is a complete discretely valued field that is either finite or has a finite residue field. Local fields play a crucial role in number theory as they provide a framework to study properties of numbers in localized settings, allowing for techniques such as completion and the analysis of primes in extensions.
Number theoretic properties: Number theoretic properties refer to the various characteristics and behaviors of numbers within the framework of number theory, particularly how they relate to divisibility, primality, congruences, and the structure of algebraic numbers. These properties help in understanding the arithmetic and algebraic structure of number fields, as well as their completions, influencing many fundamental results in algebraic number theory.
Ostrowski's Theorem: Ostrowski's Theorem states that every non-archimedean absolute value on a number field is equivalent to either a discrete valuation or the p-adic valuation for some prime p. This theorem connects the study of valuations, completions of number fields, and the structure of local fields. It plays a crucial role in understanding how number fields can be completed and analyzed through their valuations.
P-adic completion: p-adic completion is a process that creates a complete metric space from the field of rational numbers by introducing a new topology based on a prime number p. This completion allows for the study of number fields by providing a way to analyze numbers in a manner that reflects their divisibility properties relative to p, leading to insights into arithmetic properties and algebraic structures.
Real completion: Real completion is a process that takes a given number field and creates a new field that contains limits of all Cauchy sequences of real numbers, making it complete with respect to the usual metric. This concept is essential in understanding how number fields can be extended to include limits of sequences that may not converge within the original field. Real completion helps in analyzing properties like convergence and continuity in the context of algebraic number theory.
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.
Valuation: A valuation is a function that assigns a non-negative real number to elements of a number field, providing a way to measure the size or 'distance' of those elements from zero. This concept is crucial as it connects various aspects of number theory, including understanding the structure of number fields, the behavior of units, the principles of approximation, and the process of completing number fields. Valuations help in determining properties such as divisibility and congruences within these mathematical constructs.