Local fields

Definition

Local fields are a class of fields that are complete with respect to a discrete valuation and have finite residue fields. They play a crucial role in number theory and algebraic geometry, especially when examining properties of schemes over different completions, which allows for the study of rational points and the behavior of varieties over various bases. Their structure enables connections to Néron models, the Hasse principle, weak approximation, global class field theory, and p-adic numbers.

5 Must Know Facts For Your Next Test

1. Local fields include the p-adic numbers and finite extensions of them, making them essential for understanding number theoretic properties.
2. The residue field of a local field is always finite, which means it has a finite number of elements, influencing the way we study algebraic equations.
3. Néron models provide a framework to understand how abelian varieties behave over local fields, which is crucial for studying their reduction modulo primes.
4. Weak approximation is fundamentally connected to local fields as it examines how rational points can be approximated by local solutions.
5. Local fields facilitate the understanding of the Hasse principle by examining when global solutions can be determined from local data.

Review Questions

• How do local fields contribute to our understanding of Néron models and their applications?
  • Local fields play a pivotal role in the study of Néron models, as these models aim to understand the behavior of abelian varieties over different completions. When analyzing an abelian variety at a local field, one can assess how it reduces modulo primes. This reduction helps in establishing connections between the geometry of the variety and arithmetic properties, leading to significant insights in both number theory and algebraic geometry.
• In what ways does the Hasse principle utilize properties of local fields?
  • The Hasse principle states that a solution to a polynomial equation exists globally if it exists locally in all completions, including those at local fields. By leveraging the unique features of local fields—specifically their finite residue fields—one can determine whether solutions exist in specific places. This principle is powerful for checking rational points on varieties because local field properties make it easier to analyze local solutions effectively.
• Evaluate the implications of weak approximation in relation to local fields and rational points on algebraic varieties.
  • Weak approximation asserts that if a variety has rational points locally everywhere (including at all local fields), then it should also have a rational point globally. This relies heavily on the structure provided by local fields, as their finite residue fields and discrete valuations create an environment where rational solutions can be analyzed. Understanding this relationship allows mathematicians to make broader conclusions about the existence of solutions on varieties, thus connecting local and global properties seamlessly.

Related terms

Discrete Valuation: A function that assigns to each non-zero element of a field a non-negative integer representing its 'size' or 'order', providing a way to measure how far elements are from being invertible.

Complete Field: A field in which every Cauchy sequence converges, allowing for the extension of various analytic techniques in number theory and algebra.

Adèle Ring: A mathematical construction that provides a way to handle the local-global principles by integrating local fields into a single structure that captures both local and global properties.