🔢Algebraic Number Theory Unit 5 – Integral Bases & Dedekind Domains

Key Concepts and Definitions

• Integral basis a linearly independent set of elements in an algebraic number field that generates the ring of integers as a $\mathbb{Z}$-module
• Dedekind domain an integral domain in which every nonzero proper ideal factors into a product of prime ideals
  • Equivalently, an integral domain that is Noetherian, integrally closed, and has Krull dimension 1
• Fractional ideal a subset of a field that is closed under addition and multiplication by elements of the ring of integers
• Ideal class group the group of fractional ideals modulo the subgroup of principal fractional ideals
• Norm of an ideal the index of the ideal in the ring of integers, a generalization of the norm of an element
• Ramification the phenomenon where a prime ideal in a base ring may factor into a product of prime ideals with higher exponents in an extension ring

Integral Bases: Foundation and Importance

• Every algebraic number field $K$ has an integral basis, i.e., a basis of the ring of integers $\mathcal{O}_K$ as a $\mathbb{Z}$-module
• The existence of an integral basis is a consequence of $\mathcal{O}_K$ being a finitely generated $\mathbb{Z}$-module
• An integral basis allows for efficient representation and computation with algebraic integers
• The discriminant of an algebraic number field can be computed from the integral basis and measures the "density" of $\mathcal{O}_K$ in $K$
• Finding an integral basis is a fundamental problem in computational algebraic number theory
  • Algorithms such as the Round-Two algorithm or the LLL algorithm can be used to compute an integral basis
• The choice of integral basis is not unique, but different bases are related by unimodular transformations over $\mathbb{Z}$
• Integral bases play a crucial role in the study of factorization and ideal theory in algebraic number fields

Properties of Dedekind Domains

• Dedekind domains are a generalization of the ring of integers in an algebraic number field and share many of its desirable properties
• Every nonzero ideal in a Dedekind domain is uniquely expressible as a product of prime ideals
  • This unique factorization of ideals is analogous to the unique factorization of elements in a UFD
• The localization of a Dedekind domain at any nonzero prime ideal is a discrete valuation ring
• Dedekind domains are Noetherian, meaning every ideal is finitely generated
• Dedekind domains are integrally closed, i.e., they contain all elements of their fraction field that are roots of monic polynomials over the domain
• The Krull dimension of a Dedekind domain is 1, meaning every nonzero prime ideal is maximal
• Examples of Dedekind domains include the ring of integers $\mathbb{Z}$, the ring of integers in an algebraic number field, and the coordinate ring of a nonsingular affine curve over a field

Ideal Theory in Dedekind Domains

• The ideal class group of a Dedekind domain $R$, denoted $\text{Cl}(R)$, measures the failure of unique factorization of elements in $R$
  • $\text{Cl}(R)$ is trivial if and only if $R$ is a principal ideal domain (PID)
• The ideal class group is a finite abelian group, and its order is called the class number of $R$
• The class number of the ring of integers in an algebraic number field $K$ is a measure of the arithmetic complexity of $K$
• The ideal class group can be computed using the Minkowski bound and the Hermite Normal Form of the coefficient matrix of the defining equations of ideals
• The Picard group $\text{Pic}(R)$ of a Dedekind domain $R$ is isomorphic to the ideal class group $\text{Cl}(R)$
• The ideal class group plays a central role in the study of the Hilbert class field, the maximal unramified abelian extension of an algebraic number field

Factorization and Prime Ideals

• In a Dedekind domain, every nonzero proper ideal has a unique factorization as a product of prime ideals
• Prime ideals in the ring of integers $\mathcal{O}_K$ of an algebraic number field $K$ lie above prime ideals in $\mathbb{Z}$
  • The lying above relation is characterized by the splitting of primes in the extension $K/\mathbb{Q}$
• The Dedekind-Kummer theorem relates the factorization of prime ideals in an extension of Dedekind domains to the factorization of the minimal polynomial modulo the prime ideal in the base domain
• The splitting type of a prime ideal $\mathfrak{p}$ in an extension $L/K$ is determined by the degrees of the prime factors of $\mathfrak{p}\mathcal{O}_L$
  • Possible splitting types include inert, split, and ramified primes
• The Chebotarev density theorem describes the density of primes with a given splitting type in a Galois extension
• Factorization of ideals in Dedekind domains is closely related to the decomposition and inertia groups in Galois theory

Applications to Algebraic Number Fields

• The ring of integers $\mathcal{O}_K$ of an algebraic number field $K$ is a Dedekind domain, allowing the application of ideal theory to study arithmetic properties of $K$
• The class number formula relates the class number of $\mathcal{O}_K$ to the regulator, discriminant, and Dedekind zeta function of $K$
• The unit theorem describes the structure of the unit group $\mathcal{O}_K^{\times}$ as a finitely generated abelian group
• The Dirichlet unit theorem generalizes the unit theorem to the $S$-unit group for a finite set of primes $S$
• Ideal theory is used in the construction of the Hilbert class field and ray class fields of an algebraic number field
• The study of prime ideals and their density is crucial in the development of class field theory and the Langlands program
• Dedekind domains and their ideal theory have applications in cryptography, such as in the construction of ideal lattices and the study of elliptic curves over finite fields

Computational Techniques and Examples

• Computing an integral basis for an algebraic number field can be done using the Round-Two algorithm or the LLL algorithm
  • These algorithms rely on finding a reduced basis for a lattice associated with the field
• The Hermite Normal Form of the coefficient matrix of the defining equations of ideals can be used to compute the ideal class group
• The Minkowski bound provides an upper bound for the norm of an ideal in a given ideal class, aiding in the computation of the class group
• Examples of Dedekind domains and their class groups:
  • $\mathbb{Z}[\sqrt{-5}]$: class number 2, class group isomorphic to $\mathbb{Z}/2\mathbb{Z}$
  • $\mathbb{Z}[\sqrt{-23}]$: class number 3, class group isomorphic to $\mathbb{Z}/3\mathbb{Z}$
• Computing the unit group and solving the Pell equation can be done using the continued fraction algorithm or the Dirichlet unit theorem
• Symbolic computation software such as Magma, Pari/GP, and SageMath provide functions for working with Dedekind domains, ideal factorization, and class groups

Connections to Other Areas of Mathematics

• Dedekind domains and their ideal theory are closely related to the study of algebraic curves and algebraic geometry
  • The coordinate ring of a nonsingular affine curve over a field is a Dedekind domain
• The ideal class group of a Dedekind domain is isomorphic to the Picard group, which classifies isomorphism classes of invertible sheaves on the associated scheme
• The study of Dedekind domains and their completions leads to the theory of valuation rings and Prüfer domains in commutative algebra
• The Dedekind zeta function of an algebraic number field is a generalization of the Riemann zeta function and has deep connections to analytic number theory and the Langlands program
• Ideal lattices, constructed from ideals in Dedekind domains, have applications in cryptography and the study of sphere packings
• The theory of Dedekind domains and their ideal class groups is a foundation for the development of class field theory and the study of abelian extensions of algebraic number fields
• The study of factorization in Dedekind domains is related to the theory of Galois groups and the Chebotarev density theorem in algebraic number theory