12.2 Galois theory in algebraic number theory

Galois theory in algebraic number theory unveils the hidden symmetries of number fields. It connects field extensions to group structures, offering a powerful toolkit for analyzing prime factorization, ideal class groups, and field properties.

This framework illuminates the arithmetic of algebraic integers and prime splitting. By linking Galois groups to field properties, it enables deeper insights into number fields and paves the way for advanced results in class field theory.

Galois theory in number fields

Fundamentals of Galois theory in algebraic number fields

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• Galois theory offers a robust framework for analyzing symmetries and structural properties of algebraic number fields, which are finite extensions of the rational numbers Q
• The Galois group of an algebraic number field captures essential information about the field, including its degree, subfields, and prime splitting behavior
• Galois theory enables the classification of algebraic number fields based on their Galois groups, giving rise to important concepts such as Galois extensions, abelian extensions, and solvable extensions
• The Galois correspondence establishes a one-to-one correspondence between the subfields of an algebraic number field and the subgroups of its Galois group, revealing a profound connection between field theory and group theory

Applications of Galois theory to arithmetic properties

• Galois theory facilitates the study of arithmetic properties of algebraic number fields, such as the structure of their ring of integers and ideal factorization
• The ring of integers of an algebraic number field is a fundamental object of study, and Galois theory provides insights into its properties and symmetries
• Ideal factorization in algebraic number fields is a central topic in algebraic number theory, and Galois theory offers powerful tools for understanding the splitting behavior of ideals
• Galois theory can be used to analyze the class group of an algebraic number field, which measures the failure of unique factorization in its ring of integers

Galois theory for algebraic integers

Properties of algebraic integers under Galois action

• Galois theory can be applied to investigate the properties of algebraic integers, which are elements of an algebraic number field that are roots of monic polynomials with integer coefficients
• The Galois group of an algebraic number field acts on its ring of integers, and the fixed points under this action are precisely the rational integers (Z)
• The action of the Galois group on algebraic integers reveals important structural and symmetry properties of the ring of integers
• Studying the Galois action on algebraic integers leads to a deeper understanding of the arithmetic and algebraic properties of number fields

Galois theory and ideal class groups

• The ideal class group of an algebraic number field quantifies the failure of unique factorization in its ring of integers, and Galois theory provides tools for computing and analyzing this group
• The Hilbert class field of an algebraic number field, which is its maximal unramified abelian extension, has a Galois group isomorphic to the ideal class group of the field
• Galois cohomology can be employed to study the structure of ideal class groups and their relationships with other arithmetic invariants of algebraic number fields
• The connection between Galois groups and ideal class groups is a fundamental aspect of algebraic number theory and has far-reaching consequences in the study of number fields

Galois groups and prime ideal splitting

Relationship between Galois groups and prime splitting

• The splitting behavior of prime ideals in an algebraic number field extension is intimately connected to the structure of its Galois group
• When a prime ideal of the base field (usually Q) is extended to an algebraic number field, it may remain prime, split into distinct prime ideals, or ramify, depending on the Galois group of the extension
• The Dedekind-Kummer theorem describes the splitting of primes in a Galois extension of algebraic number fields in terms of the factorization of certain polynomials associated with the extension
• Understanding the connection between Galois groups and prime splitting is crucial for studying the arithmetic properties of algebraic number fields

Frobenius elements and Chebotarev density theorem

• The Frobenius element of a prime ideal in a Galois extension encodes information about the splitting behavior of the prime and generates a conjugacy class in the Galois group
• The conjugacy class of the Frobenius element in the Galois group provides a way to classify prime ideals based on their splitting behavior
• The Chebotarev density theorem is a powerful generalization of Dirichlet's theorem on primes in arithmetic progressions, relating the density of primes with a given splitting behavior to the size of the corresponding conjugacy class in the Galois group
• The Chebotarev density theorem has numerous applications in algebraic number theory, including the study of prime distribution and the construction of number fields with prescribed properties

Galois theory in algebraic number theory results

Kronecker-Weber theorem and cyclotomic fields

• The Kronecker-Weber theorem asserts that every abelian extension of Q is contained in a cyclotomic field, which is generated by roots of unity, and Galois theory plays a pivotal role in its proof
• The proof of the Kronecker-Weber theorem utilizes the fact that the Galois group of a cyclotomic field is isomorphic to the multiplicative group of integers modulo the order of the root of unity (Z/nZ)^×
• Galois theory is employed to demonstrate that every subfield of a cyclotomic field is itself a cyclotomic field, corresponding to a subgroup of the Galois group
• The Kronecker-Weber theorem is a cornerstone result in the theory of abelian extensions and has significant implications for the study of cyclotomic fields and their subfields

Class field theory and conductor-discriminant formula

• Class field theory is a major branch of algebraic number theory that describes the abelian extensions of an algebraic number field in terms of its ideal class group and idele class group, heavily relying on Galois-theoretic techniques
• The conductor-discriminant formula, which relates the conductor of an abelian extension to the discriminant of its ring of integers, is another important result in algebraic number theory that depends on Galois theory
• Galois theory provides the necessary tools and framework for proving key results in class field theory, such as the existence and uniqueness of class fields and the Artin reciprocity law
• The interplay between Galois theory and class field theory has led to significant advances in our understanding of abelian extensions and their arithmetic properties