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Algebraic Number Theory
Table of Contents
Unit 11 Overview: Local Fields and Completions
11.1 p-adic numbers and fields
11.2 Completions of number fields
11.3 Local-global principle

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

  • 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 absolute value, 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 completion of a number field with respect to a prime ideal local field
  • 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
  • Ring of integers of local field discrete valuation 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 finite extension 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 Hensel's lemma
  • 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