Local class field theory

Definition

Local class field theory is a branch of number theory that investigates the abelian extensions of local fields and establishes a connection between the arithmetic of these fields and their Galois groups. It provides powerful tools for understanding the behavior of extensions and decompositions in local settings, particularly in the study of ramification and inertia, making it essential for analyzing how primes split in extensions of local fields.

5 Must Know Facts For Your Next Test

1. Local class field theory provides a way to describe abelian extensions using local Artin reciprocity, which relates characters of the Galois group to units in the local field.
2. The theory reveals how primes in the base field decompose into products of prime ideals in extensions, helping in understanding ramification and inertia.
3. There are specific local reciprocity laws that express how different local fields behave when considered under various primes, linking their Galois groups to their respective residue fields.
4. Local class field theory has profound implications for number theoretic problems, including those involving Diophantine equations and modular forms.
5. It serves as a foundational tool for global class field theory by providing insight into local behaviors that can be pieced together for global results.

Review Questions

• How does local class field theory connect the concepts of ramification and inertia in the study of local fields?
  • Local class field theory intricately links ramification and inertia by showing how these concepts interact within local fields. Ramification concerns how primes split in extensions, while inertia describes the behavior of primes that remain unramified. The inertia group acts on these unramified extensions, and through local reciprocity laws, one can analyze how these groups relate to the Galois groups, revealing a deeper understanding of how primes decompose in various extensions.
• Discuss the role of Artin reciprocity in local class field theory and its implications for understanding abelian extensions.
  • Artin reciprocity plays a central role in local class field theory by establishing a correspondence between abelian extensions of a local field and characters of its Galois group. This connection enables mathematicians to classify and analyze these extensions systematically. The implications are significant as it allows for an exploration of the structure of units in local fields, leading to insights into how abelian extensions behave under different valuations and aiding in solving various problems in number theory.
• Evaluate how local class field theory enhances our understanding of prime decomposition and its relevance in broader number theoretic contexts.
  • Local class field theory greatly enhances our understanding of prime decomposition by providing a framework for studying how primes behave in various extensions of local fields. By relating prime ideals' behavior to the structure of their Galois groups, it enables mathematicians to predict how primes will split or remain inert. This understanding is relevant not only for local considerations but also has global implications; it helps inform strategies for tackling problems involving Diophantine equations, elliptic curves, and other areas where prime behavior is crucial to solutions.

Related terms

Local field: A complete field with respect to a discrete valuation, equipped with a topology that makes it locally compact and totally disconnected.

Abelian extension: An extension of a field where the Galois group is an abelian group, meaning the elements commute under the group operation.

Inertia group: A subgroup of the Galois group that measures the unramified extensions at a prime, capturing the behavior of primes under local extensions.