Local Field

Definition

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.

5 Must Know Facts For Your Next Test

1. Local fields can be either archimedean (like the real numbers) or non-archimedean (like the p-adic numbers), with the latter being more common in number theory.
2. The structure of a local field allows for unique factorization of ideals and primes, which is fundamental for computing class numbers and analyzing ramification.
3. In local fields, every prime ideal is maximal, and this leads to simplifications when studying extensions and their properties.
4. Local fields are used extensively in class field theory to establish connections between global fields and their local behavior at various primes.
5. The completion of a number field at a prime gives rise to a local field, allowing us to analyze local properties that can then be lifted back to the global setting.

Review Questions

• How do local fields facilitate the computation of class numbers and what role does unique factorization play in this process?
  • Local fields enable the computation of class numbers through their property of unique factorization of ideals. Since every prime ideal in a local field is maximal, it simplifies the structure of the ideal class group, making it easier to analyze. By studying local behavior at various primes, we can gain insights into the overall class number of a global field.
• Discuss the implications of the Strong Approximation Theorem in the context of local fields and their role in number theory.
  • The Strong Approximation Theorem states that for any finite set of places (including both finite and infinite ones), there exists a way to approximate elements in a number field using elements from its local completions. This theorem has significant implications as it allows us to lift solutions from local fields back to global fields. By ensuring that we can solve congruences locally at each place, we can construct global solutions and understand the distribution of primes across different extensions.
• Evaluate how the concept of ramification groups enhances our understanding of local fields and their behavior under extensions.
  • Ramification groups provide deep insights into how primes behave in extensions, particularly in local fields. They help categorize how much a prime ideal splits or ramifies in an extension, leading to rich structures that can be analyzed. By examining these groups within local fields, we can derive important information about higher ramification theory and its applications in class field theory, which links local properties back to global phenomena.

Related terms

Discretely Valued Field: A field with a valuation that assigns a non-negative integer to each non-zero element, measuring the 'size' of the element in terms of a fixed uniformizer.

Completion: The process of constructing a complete space from a given metric space by filling in 'gaps' so that every Cauchy sequence converges.

Valuation Ring: A subring of a field that consists of elements with non-negative valuation, playing an essential role in the structure of local fields.