2.2 Ring of integers and integral basis
4 min read•july 30, 2024
Number fields expand on rational numbers, introducing algebraic elements. The in these fields, denoted , consists of all within the field. This concept generalizes regular integers to more complex number systems.
O_K forms a crucial structure in algebraic number theory. It's a subring of the number field, finitely generated as a . An for O_K allows representation of all its elements, facilitating calculations and deeper theoretical insights.
Ring of Integers in Number Fields
Definition and Characteristics
Top images from around the web for Definition and Characteristics
number theory - ALGORITHM Multiplication of Integers from "discrete math and its applications ... View original
Is this image relevant?
Quadratic integer - Wikipedia, the free encyclopedia View original
Is this image relevant?
abstract algebra - Why are rings called rings? - Mathematics Stack Exchange View original
Is this image relevant?
number theory - ALGORITHM Multiplication of Integers from "discrete math and its applications ... View original
Is this image relevant?
Quadratic integer - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Characteristics
number theory - ALGORITHM Multiplication of Integers from "discrete math and its applications ... View original
Is this image relevant?
Quadratic integer - Wikipedia, the free encyclopedia View original
Is this image relevant?
abstract algebra - Why are rings called rings? - Mathematics Stack Exchange View original
Is this image relevant?
number theory - ALGORITHM Multiplication of Integers from "discrete math and its applications ... View original
Is this image relevant?
Quadratic integer - Wikipedia, the free encyclopedia View original
Is this image relevant?
1 of 3
- Ring of integers O_K in a number field K encompasses all algebraic integers within K
- Algebraic integers constitute complex numbers serving as roots of monic polynomials with integer coefficients
- O_K forms a subring of K and an
- O_K exhibits finite generation as a Z-module, with rank matching the degree of K over Q
- of Z in K manifests as O_K, incorporating all K elements integral over Z
- Q(√d) showcase specific O_K forms contingent on d ≡ 1 (mod 4) status
- For d ≡ 1 (mod 4): O_K = Z[(1+√d)/2]
- For d ≢ 1 (mod 4): O_K = Z[√d]
- O_K plays a pivotal role in algebraic number theory, extending the integer concept to number fields
Examples and Applications
- Z[i] represent the ring of integers for Q(i)
- Z[ω], where ω is a cube root of unity, form the ring of integers for Q(√-3)
- Ring of integers for Q(√2) takes the form Z[√2]
- O_K facilitates the study of and in number fields
- in O_K elucidates the behavior of rational primes in number field extensions
Integral Basis for Number Fields
Fundamental Concepts
- Integral basis {ω1, ..., ωn} for number field K comprises O_K elements forming a Z-basis for O_K
- Existence of integral basis stems from O_K's status as a of rank n (n = degree of K over Q)
- Quadratic fields Q(√d) exhibit distinct integral bases based on d's congruence modulo 4
- d ≡ 1 (mod 4): Integral basis {1, (1+√d)/2}
- d ≢ 1 (mod 4): Integral basis {1, √d}
- Cubic fields often require and index formulas for integral basis determination
- Higher degree fields may necessitate advanced techniques (Round 2 algorithm, p-adic methods) for integral basis identification
- Integral basis enables representation of all O_K elements as Z-linear combinations of basis elements
- Non-uniqueness of integral basis persists, with unimodular transformations over Z relating different bases
Examples and Applications
- Q(√5) integral basis: {1, (1+√5)/2} (golden ratio appears)
- Q(√-7) integral basis: {1, √-7}
- Integral basis for Q(∛2): {1, ∛2, (∛2)²}
- Representation of elements using integral basis: α = a + b√d in Q(√d) when d ≢ 1 (mod 4)
- Integral basis facilitates norm and trace calculations in number fields
Discriminant of a Number Field
Definition and Properties
- Number field K discriminant defined as determinant of for integral basis
- Discriminant formula: det(Tr(ωiωj)) for integral basis {ω1, ..., ωn}, Tr denoting trace function from K to Q
- Discriminant remains invariant across integral basis choices
- Absolute discriminant value gauges ring of integers "size" and informs K ramification
- Quadratic fields Q(√d) discriminant determination:
- d ≡ 2,3 (mod 4): Discriminant = 4d
- d ≡ 1 (mod 4): Discriminant = d
- Number field discriminant always yields non-zero integer
- Discriminant sign correlates with K's complex embedding count
- Minkowski's bound links discriminant to existence of small-norm non-trivial O_K elements
Calculation Examples and Applications
- Q(√5) discriminant: 5
- Q(√-7) discriminant: -28
- Q(∛2) discriminant: -108
- Discriminant aids in determining integral basis for number fields
- Ramification of primes in number field extensions relates to discriminant factorization
- Class number estimation employs discriminant in its calculations
Properties of the Ring of Integers
Dedekind Domain Characteristics
- definition: Noetherian, integrally closed integral domain with maximal non-zero prime ideals
- O_K Noetherian property proof utilizes finite generation as Z-module
- O_K integral closure in fraction field K stems from definition as Z's integral closure in K
- Non-zero prime ideal maximality in O_K proven via finite field nature of prime ideal quotients
- Ascending chain condition on ideals satisfaction demonstrates O_K's Noetherian property
- of non-zero O_K ideals as prime ideal products exemplifies key Dedekind domain trait
- O_K fractional ideals form multiplication group, characteristic of Dedekind domains
Additional Properties and Examples
- O_K lacks unique factorization for elements in general (counterexample: Z[√-5])
- Ideal class group measures deviation from unique factorization in O_K
- (PIDs) form a subset of Dedekind domains (Z[i] as an example)
- O_K satisfies the Chinese Remainder Theorem for pairwise coprime ideals
- Localization of O_K at prime ideals yields discrete valuation rings
Key Terms to Review (21)
Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They play a crucial role in number theory, particularly in the study of unique factorization and properties of number fields, connecting various concepts like integral bases and prime ideals.
Class Groups: Class groups are algebraic structures that help in understanding the ideal class structure of the ring of integers in a number field. They encapsulate how ideals in the ring can fail to be principal, providing a way to classify these ideals into equivalence classes. This concept is crucial for determining the arithmetic properties of number fields and connects deeply with topics such as unique factorization and the structure of units.
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.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Eisenstein Integers: Eisenstein integers are complex numbers of the form $z = a + b\omega$, where $a$ and $b$ are integers and $\omega = \frac{-1 + \sqrt{-3}}{2}$ is a primitive cube root of unity. This set of numbers forms a unique ring that has interesting properties, particularly in relation to factorization and ideal theory, connecting deeply with the structure of rings of integers in algebraic number fields and extending concepts found in Gaussian integers.
Free Z-module: A free Z-module is a module over the ring of integers, Z, that has a basis, meaning it can be expressed as a direct sum of copies of Z. This allows for every element of the module to be uniquely represented as an integer linear combination of the basis elements, highlighting its structure and independence. The concept plays a crucial role in understanding the properties of modules and their applications in algebraic number theory, particularly in relation to the ring of integers and integral bases.
Gaussian Integers: Gaussian integers are complex numbers of the form $$a + bi$$ where both $$a$$ and $$b$$ are integers, and $$i$$ is the imaginary unit satisfying $$i^2 = -1$$. They form a unique ring that extends the concept of integers to include imaginary units, allowing for a rich structure where concepts like factorization and primality can be studied similarly to traditional integers.
Ideal Factorization: Ideal factorization refers to the process of expressing an ideal in a ring as a product of prime ideals, similar to how integers can be expressed as a product of prime numbers. This concept is crucial for understanding the structure of rings of integers and algebraic integers, where it reveals how ideals behave in relation to one another and how they can be decomposed within larger number fields or rings.
Integral Basis: An integral basis is a set of elements in a number field that serves as a $ ext{Z}$-module basis for the ring of integers of that field. It provides a way to express every element of the ring of integers as a unique $ ext{Z}$-linear combination of these basis elements, capturing the algebraic structure and relationships within the number field.
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.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors and a multiplicative identity, where the cancellation law holds. This means that in an integral domain, if the product of two elements is zero, at least one of those elements must be zero. Integral domains are essential for studying unique factorization and prime elements, as they provide a structured environment for exploring these concepts.
Minkowski Bound: The Minkowski Bound is a critical concept in algebraic number theory that provides a bound for the non-zero ideal classes in a number field. It essentially helps to determine the size of the class group, which consists of the equivalence classes of fractional ideals. The bound can be calculated using the discriminant of the number field and is instrumental in understanding the structure of the ring of integers and its integral basis, as well as facilitating calculations related to class numbers and ideal class groups.
Noetherian Ring: A Noetherian ring is a type of ring that satisfies the ascending chain condition on ideals, meaning every increasing sequence of ideals stabilizes. This concept ensures that every ideal in the ring is finitely generated, which has crucial implications for understanding structure and behavior in rings, especially in relation to integral domains and ideal operations.
O_k: In algebraic number theory, $$o_k$$ refers to the ring of integers of a number field $$k$$, which consists of all elements in the field that are integral over the integers. This concept is crucial as it allows mathematicians to study the algebraic properties of number fields and their relationships with the integers, providing a structured way to analyze divisibility, factorization, and units within these fields.
Prime Factorization: Prime factorization is the process of breaking down a composite number into a product of its prime factors. This means expressing the number as a multiplication of prime numbers, which is unique for each number, except for the order of the factors. Understanding prime factorization is key to grasping concepts like the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 either is prime itself or can be uniquely represented as a product of prime numbers. Additionally, this concept plays an important role in the ring of integers, where integral bases can be analyzed in terms of their prime constituents.
Principal Ideal Domains: A principal ideal domain (PID) is a type of integral domain in which every ideal is generated by a single element. This structure allows for unique factorization properties, as every non-zero element can be expressed as a product of irreducible elements, much like prime factorization in the integers. In a PID, the connection to unique factorization and the ring of integers reinforces the significance of its properties in number theory.
Quadratic Number Fields: Quadratic number fields are a specific type of number field that can be expressed in the form $$ ext{Q}(\sqrt{d})$$, where $$d$$ is a square-free integer. These fields arise from adjoining a square root of a non-square integer to the rational numbers, creating an extension that has unique properties, particularly in relation to their ring of integers and integral bases. Understanding these fields helps in analyzing the algebraic structure of integers within them and their applications in various areas of number theory.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Trace Matrix: A trace matrix is a mathematical construct that summarizes the traces of linear transformations associated with elements of a ring of integers. It plays a vital role in the study of integral bases and helps in understanding how algebraic integers behave under field extensions. The trace of an element can provide insight into the structure and properties of the ring, particularly when analyzing its integral basis.
Unique Factorization: Unique factorization refers to the property of integers and certain algebraic structures where every element can be expressed uniquely as a product of irreducible elements, up to ordering and units. This concept is crucial in understanding the structure of rings and fields, as it establishes a foundational aspect of number theory that extends into the realm of algebraic number theory, where unique factorization might not hold in every context.
Z-module: A z-module is an algebraic structure consisting of an abelian group equipped with a scalar multiplication by integers. This concept generalizes the idea of vector spaces, where the scalars come from a field, allowing for the study of modules over rings, particularly focusing on the ring of integers. The structure is essential in understanding how groups can interact with integers and provides a framework for integral bases in number theory.