🔢Algebraic Number Theory Unit 6 – Ideals and Factorization in Dedekind Domains

Key Concepts and Definitions

• Dedekind domain: an integral domain in which every nonzero proper ideal factors into a product of prime ideals
• Ideal: a subset of a ring that is closed under addition and multiplication by elements of the ring
  • Principal ideal: an ideal generated by a single element
  • Prime ideal: a proper ideal $P$ such that if $ab \in P$, then either $a \in P$ or $b \in P$
  • Maximal ideal: a proper ideal that is not contained in any other proper ideal
• Factorization: the process of expressing an element or ideal as a product of irreducible or prime elements or ideals
• Unique factorization: a property of certain rings where elements can be uniquely expressed as a product of irreducible factors (up to unit factors and order)
• Noetherian ring: a ring in which every ideal is finitely generated
• Integrally closed: a domain $A$ is integrally closed if, whenever an element $x$ in the field of fractions of $A$ satisfies a monic polynomial with coefficients in $A$, then $x \in A$

Historical Context and Motivation

• Dedekind introduced the concept of Dedekind domains in the late 19th century to study algebraic number fields
• The study of ideal theory arose from the need to generalize unique factorization to rings beyond the integers
• Kummer's work on cyclotomic fields and ideal numbers laid the foundation for Dedekind's theory
• Dedekind's work on ideals and unique factorization provided a powerful tool for studying algebraic number fields and their rings of integers
• The concept of Dedekind domains has since found applications in various areas of mathematics, including algebraic geometry and commutative algebra
• Dedekind's work on ideals and unique factorization has had a profound impact on the development of modern abstract algebra and number theory

Properties of Dedekind Domains

• Every Dedekind domain is a Noetherian ring, which means that every ideal is finitely generated
• Dedekind domains are integrally closed, meaning that if an element in the field of fractions satisfies a monic polynomial with coefficients in the domain, then the element is in the domain
• The localization of a Dedekind domain at any nonzero prime ideal is a discrete valuation ring
• The group of fractional ideals of a Dedekind domain forms a group under multiplication, with the principal fractional ideals forming a subgroup
• The class group of a Dedekind domain, which measures the failure of unique factorization, is the quotient of the group of fractional ideals by the subgroup of principal fractional ideals
• Every nonzero ideal in a Dedekind domain has a unique factorization as a product of prime ideals

Ideals in Dedekind Domains

• Every nonzero proper ideal in a Dedekind domain can be uniquely factored as a product of prime ideals
• The prime ideals in a Dedekind domain satisfy the following properties:
  • Every nonzero prime ideal is maximal
  • The intersection of all nonzero prime ideals is zero
  • Every nonzero ideal is contained in only finitely many prime ideals
• The prime ideals in the ring of integers of an algebraic number field are closely related to the prime numbers that split, remain inert, or ramify in the field extension
• The decomposition of prime ideals in extensions of Dedekind domains is described by the Lying Over, Going Up, and Going Down theorems
• The Chinese Remainder Theorem for Dedekind domains states that if $I_1, \ldots, I_n$ are pairwise coprime ideals, then the ring modulo their product is isomorphic to the product of the rings modulo each ideal

Prime Factorization Theorem

• The Prime Factorization Theorem for Dedekind domains states that every nonzero proper ideal in a Dedekind domain can be uniquely factored as a product of prime ideals
• The factorization is unique up to the order of the prime ideals
• The theorem generalizes the Fundamental Theorem of Arithmetic for the integers to the setting of Dedekind domains
• The prime factorization of an ideal can be computed using the decomposition of prime ideals in extensions of Dedekind domains
• The prime factorization of an ideal is closely related to the factorization of the norm of the ideal in the integers
• The prime factorization of ideals plays a crucial role in the study of class groups and the structure of the group of fractional ideals in a Dedekind domain

Applications in Number Theory

• Dedekind domains and their ideal theory have numerous applications in algebraic number theory
• The ring of integers of an algebraic number field is a Dedekind domain, and the prime ideals correspond to the prime numbers that split, remain inert, or ramify in the field extension
• The class group of an algebraic number field, which measures the failure of unique factorization, can be studied using the ideal theory of Dedekind domains
• The prime factorization of ideals is used to compute the decomposition of prime numbers in algebraic number fields
• Dedekind domains and their ideal theory are used to study the arithmetic of elliptic curves and other algebraic varieties
• The study of Dedekind domains and their class groups has led to important results in class field theory and the Langlands program

Computational Techniques

• Computing the prime factorization of an ideal in a Dedekind domain can be done using various algorithms, such as the Round 2 algorithm or the Buchmann-Lenstra algorithm
• The computation of the class group of a Dedekind domain can be done using algorithms such as the Hafner-McCurley algorithm or the Bach-Sorenson algorithm
• The computation of the unit group of a Dedekind domain (the group of invertible elements) can be done using algorithms such as the Buchmann-Pethő algorithm or the Simon-Pauli algorithm
• Efficient algorithms for ideal arithmetic (addition, multiplication, and division) in Dedekind domains are essential for computational applications
• The LLL algorithm can be used to find short vectors in lattices associated with ideals in Dedekind domains, which has applications in cryptography and Diophantine approximation
• Computational techniques for Dedekind domains often rely on the relationship between ideals and lattices, as well as the geometry of numbers

Advanced Topics and Extensions

• The study of Dedekind domains can be generalized to the setting of Dedekind rings, which are rings where every localization at a maximal ideal is a discrete valuation ring
• The theory of Dedekind domains can be extended to the setting of function fields, where the role of prime numbers is played by places of the function field
• The study of Dedekind domains and their class groups is closely related to the study of abelian extensions of number fields and class field theory
• The Hilbert class field of a number field is the maximal unramified abelian extension of the field, and its Galois group is isomorphic to the class group of the ring of integers
• The Chebotarev Density Theorem describes the distribution of prime ideals in Galois extensions of number fields, generalizing the Prime Number Theorem
• The study of Dedekind domains and their ideal theory has connections to the Langlands program, which seeks to unify various areas of mathematics, including number theory, representation theory, and harmonic analysis