Local field

Definition

A local field is a field that is complete with respect to a discrete valuation and has a finite residue field. This concept plays a crucial role in various areas of number theory and algebraic geometry, serving as a foundational building block for the study of local properties of schemes and arithmetic objects. Local fields provide a framework for understanding the behavior of algebraic varieties over both finite and infinite extensions, especially in relation to their rigid analytic structures and class field theory.

5 Must Know Facts For Your Next Test

1. Local fields can be either finite extensions of the rational numbers or function fields over finite fields, including examples such as the p-adic numbers and function fields in one variable over a finite field.
2. The topology on local fields is induced by the valuation, allowing for convergence and limits to be defined in a way that mirrors familiar calculus concepts.
3. Local fields are critical in the formulation of local class field theory, which explores how abelian extensions correspond to Galois groups in the context of number theory.
4. They are instrumental in the study of rigid analytic spaces, where local fields provide the underlying algebraic structures needed to understand analytic properties.
5. The concept of local fields connects deeply with global fields; many results in number theory can be understood better through their local counterparts using local fields.

Review Questions

• How does the concept of a local field enhance our understanding of rigid analytic spaces?
  • Local fields are essential for understanding rigid analytic spaces because they allow us to apply concepts from p-adic analysis to study algebraic varieties. The topology of local fields, defined by discrete valuations, enables us to analyze convergence and continuity within these spaces. This connection helps in exploring properties like analyticity, which parallels classical complex analysis but is tailored for the context of algebraic varieties over local fields.
• In what ways does local class field theory utilize the structure of local fields to relate abelian extensions with Galois groups?
  • Local class field theory leverages local fields to create a bridge between abelian extensions and Galois groups through the use of reciprocity laws. By focusing on local fields, we can develop explicit constructions for maximal abelian extensions, which correspond directly to Galois groups. This relationship enhances our understanding of both algebraic number theory and the behavior of various arithmetic objects under local conditions.
• Discuss how the interplay between local and global fields deepens our comprehension of number theoretic phenomena.
  • The relationship between local and global fields reveals profound insights into number theoretic phenomena through the principle of local-global compatibility. By examining local properties via local fields—like completions and discrete valuations—we can infer global behaviors concerning rational points and divisibility. This interplay is crucial for understanding solutions to Diophantine equations and allows mathematicians to apply tools from p-adic analysis to solve problems that arise in the broader context of algebraic number theory.

Related terms

Discrete Valuation:

A valuation on a field that assigns to each non-zero element a unique non-negative integer, providing a way to measure 'size' or 'order' of elements.

Residue Field:

The field obtained by taking the quotient of a local field by its maximal ideal, capturing the 'simplest' elements of the local field's structure.

Rigid Analytic Space: A type of space that allows for the study of analytic geometry over local fields, focusing on properties similar to those in classical complex analysis.