5.2 Dedekind domains: definition and characteristics
3 min read•july 30, 2024
Dedekind domains are special rings in algebra that behave nicely with prime ideals. They're like the cool kids of number theory, making factorization and ideal arithmetic work smoothly. You'll love how they simplify complex ideas!
In this part of the course, we're looking at how Dedekind domains connect to number fields and their rings of integers. It's all about understanding the structure of these rings and how they help us solve tricky number theory problems.
Dedekind Domains
Definition and Key Properties
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- integral domain , integrally closed, every non-zero prime ideal maximal
- Noetherian rings characterized by ascending chain condition on ideals ensures every ideal finitely generated
- Integral domain R integrally closed contains all elements of field of fractions roots of monic polynomials with coefficients in R
- Non-zero ideal uniquely factored as product of prime ideals, up to order of factors
- Satisfy cancellation law for ideals for non-zero ideals A, B, and C, then
- Localization at any prime ideal discrete valuation ring
- Krull dimension 1 means any chain of prime ideals has length at most 1
Examples and Applications
- Ring of integers in algebraic number fields ()
- Coordinate ring of smooth affine algebraic curve over field
- Polynomial ring in one variable over field
- Discrete valuation rings (DVRs)
- Used in algebraic number theory to study ideal factorization and class groups
- Applied in algebraic geometry for local study of curves
Integers in Number Fields
Number Fields and Rings of Integers
- Number field K finite algebraic extension of rational numbers Q (, )
- Ring of integers elements of K roots of monic polynomials with integer coefficients
- Dedekind domain proved by establishing Noetherian, integrally closed, every non-zero prime ideal maximal
- Noetherian finitely generated Z-module follows from existence of integral basis for number field
- of in K itself proves integrally closed in field of fractions K
Proof Techniques and Key Concepts
- Non-zero prime ideal maximal shown using finite integral domain for any non-zero prime ideal P, thus field
- Proof involves using norm of ideal and properties of algebraic number theory to establish characteristics
- Integral basis concept crucial in proving finitely generated Z-module
- Discriminant of number field used to study properties of
- Minkowski's bound employed to prove finiteness of class number
Ideal Structure of Dedekind Domains
Unique Factorization of Ideals
- Non-zero ideal uniquely factored as product of prime ideals analogous to unique factorization of elements in UFD
- Fractional ideals form group under multiplication, ring itself identity element
- Class group quotient of group of fractional ideals by subgroup of principal fractional ideals measures how far domain from being principal ideal domain
- Every ideal invertible means for non-zero ideal A, exists ideal B such that AB principal
Theorems and Applications
- Chinese Remainder Theorem holds allows solution of simultaneous congruences modulo relatively prime ideals
- Approximation Theorem states for finite set of prime ideals and integers, exists element of domain with prescribed valuations at these primes
- Ideal structure leads to applications in algebraic number theory including study of ramification and decomposition of primes in number field extensions
- Used to analyze splitting of primes in field extensions
- Crucial in studying of number fields
Dedekind Domains vs Unique Factorization
Generalizations and Comparisons
- Dedekind domains generalize principal ideal domains (PIDs) and unique factorization domains (UFDs) preserves many important properties
- Every PID Dedekind domain, not every Dedekind domain PID (ring of integers in number field may fail to be PID)
- Unique factorization of elements may fail in Dedekind domain, but unique factorization of ideals into prime ideals always holds
- Failure of unique factorization of elements measured by class group Dedekind domain UFD if and only if class group trivial
Implications and Advanced Concepts
- Ideal class groups in Dedekind domains provide way to study how far domain from being UFD leads to important results in algebraic number theory
- Prime elements correspond precisely to prime principal ideals, not all prime ideals necessarily principal
- Relationship between Dedekind domains and UFDs highlights importance of studying ideal theory as generalization of element-wise factorization in more general settings
- Used to study class field theory and reciprocity laws
- Provides framework for analyzing Diophantine equations in number fields
Key Terms to Review (18)
Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. These numbers include all integers and roots of unity, forming a crucial part of algebraic number theory, particularly when discussing the properties of number fields, norms, traces, minimal polynomials, and the structure of Dedekind domains.
Arithmetic Geometry: Arithmetic geometry is a branch of mathematics that combines algebraic geometry and number theory to study solutions to polynomial equations with an emphasis on their arithmetic properties. It focuses on understanding the relationships between geometric objects defined over number fields or arithmetic schemes and the associated number-theoretic questions, bridging the gap between algebraic structures and geometric intuition.
Class Number Formula: The class number formula relates the class number of a number field to its Dedekind zeta function and other invariants like the regulator and the discriminant. This formula serves as a bridge between algebraic number theory and analytic number theory, revealing deep connections between arithmetic properties of number fields and their behavior in the complex plane.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dedekind domain: A Dedekind domain is a type of integral domain in which every non-zero proper ideal can be uniquely factored into a product of prime ideals. This property allows Dedekind domains to generalize many familiar concepts in number theory, such as the ring of integers and unique factorization, while also providing a framework for understanding fractional ideals and ideal class groups.
Dedekind Ring: A Dedekind ring is an integral domain in which every nonzero proper ideal can be factored into a product of prime ideals. This concept is essential for understanding the structure of rings and their ideals, particularly in the context of algebraic number theory, where Dedekind domains, a specific type of Dedekind ring, exhibit nice properties like the existence of unique factorization of ideals.
Every nonzero prime ideal is maximal: This statement means that in a Dedekind domain, every nonzero prime ideal is also a maximal ideal. This is a significant property of Dedekind domains, which are integral domains where every nonzero prime ideal is contained in exactly one maximal ideal. This connection reflects how the structure of ideals in Dedekind domains facilitates factorization, leading to a well-behaved arithmetic within these rings.
Field Extension: A field extension is a bigger field that contains a smaller field and allows for more solutions to polynomial equations. This concept helps in understanding how different fields relate to each other, especially when it comes to algebraic numbers, algebraic integers, and the properties of polynomials in those fields.
Fractional Ideal: A fractional ideal is a generalization of the concept of an ideal in a ring, specifically in the context of Dedekind domains. It is an additive subgroup of the field of fractions of the integral domain that can be expressed as a fractional multiple of an ideal, allowing for the treatment of elements not necessarily contained in any given ideal. This concept is essential for understanding unique factorization and the structure of the ideal class group.
Galois Theory: Galois Theory is a branch of mathematics that connects field theory and group theory, providing a framework to understand the symmetries of the roots of polynomial equations. It explores how the structure of field extensions relates to the properties of groups, especially focusing on the relationships between subfields and subgroups. This theory serves as a fundamental tool for determining when a polynomial can be solved by radicals and plays a crucial role in understanding the solvability of polynomial equations.
Ideal Class Group: The ideal class group is a fundamental concept in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It consists of equivalence classes of fractional ideals, where two ideals are considered equivalent if their product with a principal ideal is again a fractional ideal. This group plays a crucial role in understanding the structure of rings of integers and their relationship to number fields, helping to connect various areas such as discriminants, integral bases, and the properties of Dedekind domains.
Integer Ring: An integer ring is a specific type of ring in abstract algebra, consisting of the set of integers $$ extbf{Z}$$ along with the operations of addition and multiplication. This structure satisfies properties such as closure, associativity, commutativity for both operations, and the existence of additive identities and inverses. The integer ring is fundamental in understanding more complex algebraic systems, particularly as it serves as an essential example of a commutative ring with unity and is vital in the study of unique factorization and ideals.
Integral Closure: Integral closure refers to the set of all elements in a given field that are integral over a specified ring, particularly focusing on algebraic integers. It connects various concepts like algebraic numbers and integers, providing a way to understand the structure of rings of integers in number fields, ensuring that algebraic properties are preserved within extensions.
Noetherian: Noetherian refers to a type of ring that satisfies the ascending chain condition on ideals, meaning that every increasing sequence of ideals eventually stabilizes. This concept is crucial because it ensures that certain algebraic properties, like finitely generated modules having a well-defined structure, hold true. Noetherian rings are foundational in various areas of algebra, particularly in the study of Dedekind domains, as they guarantee that many useful results can be applied when analyzing the structure of these rings.
Polynomial rings over a field: A polynomial ring over a field is a mathematical structure consisting of polynomials whose coefficients come from a given field. This ring is denoted as $F[x]$, where $F$ is the field and $x$ is an indeterminate. Polynomial rings exhibit important algebraic properties, such as being a unique factorization domain (UFD) and having well-defined ideals, which play a crucial role in the study of algebraic structures like Dedekind domains.
Richard Dedekind: Richard Dedekind was a prominent German mathematician known for his contributions to abstract algebra and number theory, particularly in the development of ideals and the concept of Dedekind domains. His work laid the foundation for understanding the structure of number fields and their properties, which are central to modern algebraic number theory.
Ring of integers of a number field: The ring of integers of a number field is the integral closure of the integers in that field, serving as a generalization of the concept of integers to more complex algebraic structures. This ring consists of all elements in the number field that are roots of monic polynomials with coefficients in the integers, forming a critical structure for understanding the arithmetic properties of number fields. The ring of integers plays a key role in defining Dedekind domains, which are integral domains where every non-zero prime ideal is maximal.
Theorem of Uniqueness of Factorization: The theorem of uniqueness of factorization states that in certain algebraic structures, like integers or polynomials over a field, every element can be expressed uniquely as a product of irreducible elements, up to order and units. This concept connects deeply with the properties of Dedekind domains, which are integral domains where every non-zero prime ideal is maximal, ensuring a form of unique factorization in the context of ideals rather than elements.