12.1 Representations in algebraic number theory

Algebraic number theory uses representations to study complex number structures. These tools illuminate field properties, analyze extensions, and reveal arithmetic insights through character theory and L-functions.

Representations shine in proofs and zeta functions. They're key in the Dirichlet unit theorem and help decode arithmetic info in Dedekind zeta functions. This connects to broader themes in algebraic number theory.

Representations in Algebraic Number Theory

Representations in algebraic number fields

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• Algebraic number fields extend rational numbers finitely creating complex structures (quadratic fields, cyclotomic fields)
• Galois group representations illuminate field properties through character theory and Artin representations
• Representations analyze field extensions, ramification behavior, and arithmetic properties of number fields
• Character theory applied to finite abelian groups reveals ideal class group structure
• L-functions connect to class numbers, providing insights into field properties
• Dirichlet characters employed to study multiplicative properties of number fields

Representation theory for ideal classes

• Ideal class group measures how far a ring of integers deviates from unique factorization, always finite
• Unit group consists of invertible elements in the ring of integers, structure determined by field properties
• Character theory of finite abelian groups applied to ideal class groups reveals group structure
• L-functions relate to class numbers, providing information about ideal class group size
• Dirichlet characters used to study distribution of prime ideals in number fields

Representations in number theory proofs

• Dirichlet unit theorem states rank of unit group equals $r_1 + r_2 - 1$ ($r_1$ real embeddings, $r_2$ pairs of complex embeddings)
• Proof techniques employ character theory of finite abelian groups and Fourier analysis on finite groups
• Theorem applications:
  1. Determine structure of units in number fields
  2. Compute fundamental units
  3. Study growth of units in tower of number fields
• Representations facilitate proofs of other key results (class number formula, Chebotarev density theorem)

Representations and zeta functions

• Dedekind zeta function encodes arithmetic information of number field, admits analytic continuation and functional equation
• Artin L-functions generalize Dedekind zeta function, associated with Galois representations
• Hecke L-functions extend Dirichlet L-functions to general number fields
• Class number formula connects special values of zeta functions to important invariants of number fields
• Brauer-Siegel theorem describes asymptotic behavior of class numbers and regulators in sequences of number fields