Abelian extension

Definition

An abelian extension is a field extension of a number field that is both normal and separable, where the Galois group of the extension is an abelian group. This concept plays a crucial role in understanding the relationships between number fields and their arithmetic properties, linking to various advanced topics like cyclotomic fields and class field theory, which address the structure of these extensions and their applications in algebraic number theory.

5 Must Know Facts For Your Next Test

1. Abelian extensions can be described using Galois theory, where the structure of the Galois group provides insights into the arithmetic properties of the extension.
2. Cyclotomic fields are prime examples of abelian extensions, demonstrating how roots of unity generate these extensions and their relationship with modular forms.
3. In class field theory, every abelian extension corresponds to a certain type of ideal class group, illustrating the deep connections between field extensions and algebraic structures.
4. The notion of local and global fields relates to abelian extensions, where local field extensions can provide insight into global properties through local-global principles.
5. Abelian extensions are essential for understanding reciprocity laws in number theory, connecting them to the distribution of prime numbers in arithmetic progressions.

Review Questions

• How does the Galois group relate to the concept of an abelian extension and what implications does this have for understanding the properties of number fields?
  • The Galois group associated with an abelian extension reveals its structure by indicating how many distinct automorphisms exist that keep the base field unchanged. In an abelian extension, this Galois group is commutative, which simplifies understanding its properties and behaviors. This commutativity often leads to easier analysis of the roots of polynomials and helps in classifying extensions, making it crucial for studying number fields.
• Discuss how cyclotomic fields serve as a primary example of abelian extensions and what unique features they exhibit.
  • Cyclotomic fields are formed by adjoining a primitive $n$-th root of unity to the rationals, serving as a prime example of abelian extensions due to their well-understood structure. Their Galois groups correspond to the multiplicative group of integers modulo $n$, which is abelian. This property allows for explicit computation of class numbers and other invariants, showing how cyclotomic fields fit neatly into the broader framework of abelian extensions and class field theory.
• Evaluate how class field theory expands our understanding of abelian extensions and their relationship with ideal class groups.
  • Class field theory provides a profound link between abelian extensions and ideal class groups by establishing a correspondence that allows for the study of number fields through their ideal structures. This theory explains how every abelian extension can be represented by a specific ideal class group, thereby translating problems in field theory into questions about ideals. This connection not only enriches our understanding of both concepts but also facilitates computations in number theory related to primes and divisibility.

Related terms

Galois group: The group of automorphisms of a field extension that keeps the base field fixed, reflecting the symmetries of the roots of polynomials.

Class field theory: A branch of algebraic number theory that describes the abelian extensions of number fields in terms of ideal class groups and abelian varieties.

Cyclotomic fields: Fields obtained by adjoining a primitive $n$-th root of unity to the rationals, which are a significant example of abelian extensions.