Local Fields

Definition

Local fields are a special class of fields that are complete with respect to a discrete valuation and have finite residue fields. These fields arise in number theory and algebraic geometry, providing a framework for understanding properties of algebraic extensions, particularly in the context of inseparable extensions. The completeness and finite residue properties make local fields critical in analyzing the behavior of algebraic structures over them.

5 Must Know Facts For Your Next Test

1. Local fields can be classified into two main types: finite extensions of the field of p-adic numbers and finite extensions of function fields over a finite field.
2. Every local field has a unique maximal ideal, which corresponds to the elements that are 'small' in the sense of the discrete valuation.
3. The completion of a number field with respect to a given place gives rise to local fields, making them essential in the study of local-global principles in number theory.
4. Inseparable extensions often occur in local fields due to their characteristics, particularly when considering fields of positive characteristic.
5. Local fields allow for a more straightforward analysis of algebraic structures, as they provide a setting where convergence and limit processes are well-defined.

Review Questions

• How do local fields relate to discrete valuations, and why is this relationship important?
  • Local fields are defined through their completeness with respect to a discrete valuation, which measures the size or 'closeness' of elements. This relationship is crucial because it allows us to understand how local fields behave under various algebraic operations, providing insights into their structure. The discrete valuation also facilitates the study of convergent sequences and limits within these fields, making them useful for examining algebraic extensions.
• Discuss the significance of inseparable extensions in local fields and their implications for algebraic structures.
  • Inseparable extensions are particularly significant in local fields as they often arise when dealing with fields of positive characteristic. These extensions feature polynomials that have repeated roots, leading to distinct properties compared to separable extensions. Understanding inseparable extensions helps in analyzing the structure of local fields and provides deeper insights into their arithmetic properties, especially concerning solutions to polynomial equations.
• Evaluate the role that local fields play in connecting global properties with local behaviors in number theory.
  • Local fields serve as a bridge between global properties and local behaviors in number theory through the lens of local-global principles. By studying how arithmetic properties manifest locally, one can gain insights into global solutions to equations. This connection is vital for understanding whether certain solutions exist globally based on their existence in local contexts, ultimately impacting results like the Hasse Principle and contributing to broader discussions about rationality and solvability.

Related terms

Discrete Valuation: A function that assigns to each element of a field a non-negative integer, reflecting the 'size' of elements in relation to a chosen prime.

Inseparable Extension: An extension of fields where every element is a root of a polynomial that is not separable, typically characterized by having repeated roots.

Residue Field: The field obtained by taking the quotient of a local field by its maximal ideal, which captures the behavior of the local field at 'infinity'.