Abelian extension

Definition

An abelian extension is a field extension of a number field or local field where the Galois group of the extension is an abelian group. This type of extension is significant because it helps in understanding the relationships between different fields, particularly in relation to class field theory and the structure of Galois groups.

5 Must Know Facts For Your Next Test

1. Abelian extensions can be classified using the fundamental theorem of class field theory, which connects them to the ideal class groups.
2. For a number field, abelian extensions correspond to certain quotient groups of its Galois group, which can be interpreted through local fields.
3. The study of abelian extensions provides insights into the solvability of polynomial equations over number fields using Galois theory.
4. Abelian extensions can often be constructed using roots of unity, which play a key role in their structure and properties.
5. In local class field theory, every unramified abelian extension of a local field can be described using its maximal abelian extension.

Review Questions

• How does the concept of an abelian extension relate to the structure of Galois groups?
  • An abelian extension is directly linked to Galois groups since it specifically requires that the Galois group associated with the extension is abelian. This means that the operations within the group commute, simplifying many aspects of the analysis. Understanding this relationship allows one to use properties of abelian groups to gain insights into the behavior of field extensions and how they can be classified.
• Discuss the importance of abelian extensions in the context of class field theory.
  • Abelian extensions are central to class field theory as they provide a framework for understanding how number fields relate to their ideal class groups. The fundamental theorem establishes a correspondence between abelian extensions and these class groups, allowing mathematicians to study properties like ideal factorization and divisibility within fields. This connection highlights how abelian extensions serve as key examples for broader theories in algebraic number theory.
• Evaluate the implications of studying abelian extensions in local fields for modern number theory.
  • Studying abelian extensions in local fields has profound implications for modern number theory, especially regarding how these extensions affect arithmetic properties and their applications. The local-to-global principles that arise from this study facilitate a better understanding of Diophantine equations and their solvability across various fields. By investigating how unramified abelian extensions relate back to their maximal abelian counterparts, mathematicians can draw connections that enhance both theoretical frameworks and practical applications in number theory.

Related terms

Galois group: The group of symmetries (automorphisms) of a field extension that keeps the base field fixed, revealing information about the structure of the extension.

Class field theory: A branch of algebraic number theory that studies abelian extensions of number fields and relates them to the ideal class group of those fields.

Local field: A field that is complete with respect to a discrete valuation and has a finite residue field, often used in number theory.