Local field

Definition

A local field is a field that is complete with respect to a discrete valuation and has a finite residue field. It can be thought of as a generalization of the notion of a p-adic field or function field in one variable over a finite field. Local fields play an important role in number theory and algebraic geometry, providing a framework for understanding various concepts such as completions, valuations, and K-theory.

5 Must Know Facts For Your Next Test

1. Local fields can be classified into two main types: non-archimedean local fields, like p-adic numbers, and function fields over finite fields.
2. Every local field has a unique maximal ideal and its residue field is finite, making them suitable for applications in algebraic number theory.
3. Local fields are essential in understanding the Milnor K-theory as they allow for the examination of rationality and cohomology in a localized setting.
4. The completion of a global field at a prime leads to a local field, which means local fields can often be viewed as completions of number fields or function fields.
5. The structure of local fields facilitates the study of the Bloch-Lichtenbaum spectral sequence, linking topological properties with algebraic properties in K-theory.

Review Questions

• How does the concept of local fields relate to completions in number theory?
  • Local fields arise naturally from the process of completing global fields at specific primes. When we take a global field, such as a number field or function field, and complete it at a prime, we end up with a local field. This relationship is significant because it allows us to analyze global objects through their local properties, facilitating deeper insights into algebraic structures and their associated invariants.
• Discuss how local fields contribute to the understanding of Milnor K-theory.
  • Local fields play a crucial role in Milnor K-theory as they help in the computation of higher K-groups and their connections to Galois cohomology. The structure of local fields provides a setting where one can analyze rational functions and elements under various valuations. This understanding aids in establishing relationships between different K-groups and advancing results in both algebraic geometry and number theory.
• Evaluate the impact of local fields on the Bloch-Lichtenbaum spectral sequence and its implications for K-theory.
  • The study of local fields significantly enhances our comprehension of the Bloch-Lichtenbaum spectral sequence by providing an effective way to compute K-theory groups in localized settings. The spectral sequence links various cohomological dimensions and allows mathematicians to derive essential information about higher K-theory. Local fields facilitate this process due to their structured nature, leading to results that deepen our understanding of algebraic varieties and their properties within the framework of K-theory.

Related terms

Discrete Valuation: A function that assigns to each element of a field a non-negative integer, representing the 'order' or 'multiplicity' of that element with respect to a chosen prime.

Completion: The process of constructing a complete space from a given metric space by adding limit points, which in the context of fields refers to creating a local field from a given field.

Henselian Field: A field that satisfies Hensel's lemma, allowing for lifting roots of polynomials from the residue field to the local field itself.