Glossary

1.1 Algebraic number fields

9 min read•august 21, 2024

extend rational numbers, providing a rich structure for studying algebraic properties and solving equations. They form the foundation of arithmetic geometry, allowing us to explore advanced concepts and tackle complex mathematical problems.

These fields are constructed by adjoining roots of polynomials to the rationals. Understanding their structure, including integral elements, ideals, and class groups, is crucial for delving into the arithmetic of algebraic varieties and solving equations over number fields.

Definition of algebraic number fields

  • Algebraic number fields form the foundation of arithmetic geometry by extending the rational numbers
  • These fields provide a rich structure for studying algebraic properties and solving equations
  • Understanding algebraic number fields is crucial for exploring more advanced concepts in arithmetic geometry

Extension fields over rationals

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  • Algebraic number fields are finite extensions of the rational numbers Q\mathbb{Q}
  • Constructed by adjoining roots of polynomials with rational coefficients to Q\mathbb{Q}
  • Degree of the extension determined by the dimension as a vector space over Q\mathbb{Q}
  • Examples include:
    • Quadratic fields: Q(d)\mathbb{Q}(\sqrt{d}) where dd is a square-free integer
    • Cyclotomic fields: Q(ζn)\mathbb{Q}(\zeta_n) where ζn\zeta_n is a primitive nth root of unity

Minimal polynomials

  • Irreducible polynomial of lowest degree with rational coefficients that has a given algebraic element as a root
  • Uniquely defines an algebraic number up to isomorphism
  • Degree of the minimal polynomial equals the degree of the field extension
  • Coefficients of minimal polynomials provide important information about the algebraic number

Algebraic elements

  • Elements that are roots of non-zero polynomials with rational coefficients
  • Form the building blocks of algebraic number fields
  • Classified by their degree: the degree of their minimal polynomial
  • Transcendental numbers (non-algebraic) excluded from algebraic number fields

Field embeddings and isomorphisms

  • Field embeddings and isomorphisms play a crucial role in understanding the structure of algebraic number fields
  • These concepts allow for the comparison and classification of different number fields
  • Embeddings and isomorphisms form the basis for studying in the context of arithmetic geometry

Homomorphisms between number fields

  • Structure-preserving maps between algebraic number fields
  • Preserve addition and multiplication operations
  • Can be used to study relationships between different number fields
  • Kernel and image of homomorphisms provide insights into field structure

Automorphisms of number fields

  • Isomorphisms from a number field to itself
  • Form a group under composition (the automorphism group)
  • Fixed field of the automorphism group corresponds to the base field
  • Number of automorphisms related to the degree of the field extension

Galois theory connection

  • Galois group of a number field consists of its automorphisms
  • Fundamental theorem of Galois theory establishes a correspondence between:
    • Subfields of the number field
    • Subgroups of its Galois group
  • Galois theory used to study field extensions and solve polynomial equations

Integral elements and rings

  • Integral elements and rings form the arithmetic foundation of algebraic number fields
  • These structures generalize the concept of integers within number fields
  • Understanding integral elements is essential for studying divisibility and factorization in arithmetic geometry

Ring of integers

  • Consists of all within a number field
  • Algebraic integers defined as roots of monic polynomials with integer coefficients
  • Forms a subring of the number field
  • Generalizes the Z\mathbb{Z} in Q\mathbb{Q}

Integral basis

  • Finite set of algebraic integers that generate the entire ring of integers
  • Every element of the ring of integers can be expressed as an integer linear combination of basis elements
  • Number of elements in the equals the degree of the number field
  • Computation of integral basis can be challenging for high-degree number fields

Discriminant of number fields

  • Measures the "size" of the ring of integers
  • Defined as the determinant of the trace form on the integral basis
  • Provides information about ramification in the number field
  • Smaller absolute value of indicates "simpler" number field

Ideal theory in number fields

  • Ideal theory extends the concept of divisibility to algebraic number fields
  • This theory provides a powerful framework for studying factorization and prime decomposition
  • Understanding ideals is crucial for exploring arithmetic properties in the context of arithmetic geometry

Prime ideals

  • Proper ideals that cannot be factored into smaller non-unit ideals
  • Generalize prime numbers in the integers
  • Play a crucial role in the factorization of ideals
  • Correspond to prime elements in the ring of integers

Factorization of ideals

  • Every non-zero ideal in the ring of integers can be uniquely factored into prime ideals
  • Generalizes the fundamental theorem of arithmetic for integers
  • Factorization may not be unique for elements, but is unique for ideals
  • Provides a way to study divisibility in number fields

Dedekind domains

  • Rings of integers in algebraic number fields are Dedekind domains
  • Characterized by being Noetherian, integrally closed, and having all non-zero prime ideals maximal
  • Every non-zero fractional ideal is invertible in a
  • Dedekind domains provide a natural setting for studying ideal factorization

Norm and trace

  • Norm and trace are fundamental functions in the study of algebraic number fields
  • These concepts generalize familiar notions from linear algebra to field extensions
  • Norm and trace play crucial roles in various aspects of arithmetic geometry and number theory

Norm of elements

  • Multiplicative function that measures the "size" of elements in a number field
  • Defined as the product of all conjugates of an element
  • For a field extension K/FK/F and α∈K\alpha \in K, norm given by NK/F(α)=∏σσ(α)N_{K/F}(\alpha) = \prod_{\sigma} \sigma(\alpha)
  • Used in studying units and ideal factorization in number fields

Trace of elements

  • Additive function that provides information about elements in a number field
  • Defined as the sum of all conjugates of an element
  • For a field extension K/FK/F and α∈K\alpha \in K, trace given by TrK/F(α)=∑σσ(α)Tr_{K/F}(\alpha) = \sum_{\sigma} \sigma(\alpha)
  • Used in constructing bilinear forms and studying field discriminants

Applications in number theory

  • Norm and trace used in studying
  • Play a role in defining L-functions and zeta functions of number fields
  • Help in determining whether elements are units in the ring of integers
  • Used in the study of quadratic forms over number fields

Class group and class number

  • The class group and provide insights into the arithmetic structure of number fields
  • These concepts measure how far the ring of integers is from being a unique factorization domain
  • Understanding class groups is crucial for various applications in arithmetic geometry

Ideal class group

  • Quotient group of fractional ideals modulo principal ideals
  • Measures the failure of unique factorization in the ring of integers
  • Finite group for algebraic number fields (Minkowski's theorem)
  • Trivial class group indicates that the ring of integers is a principal ideal domain

Class number formula

  • Relates the class number to other invariants of the number field
  • Involves the regulator, discriminant, and special values of L-functions
  • For quadratic fields Q(d)\mathbb{Q}(\sqrt{d}), class number given by Dirichlet's formula
  • Generalized to arbitrary number fields using Dedekind zeta functions

Dirichlet's unit theorem

  • Describes the structure of the unit group in the ring of integers
  • States that the unit group is finitely generated
  • Rank of the unit group related to the number of real and complex embeddings
  • Fundamental units generate all units up to roots of unity

Ramification theory

  • studies how prime ideals decompose in field extensions
  • This theory provides crucial information about the arithmetic of number fields
  • Understanding ramification is essential for various aspects of arithmetic geometry

Ramified vs unramified primes

  • Prime ideal p\mathfrak{p} is ramified if it divides the discriminant of the number field
  • Unramified primes have simpler decomposition behavior
  • Ramification index measures the degree of ramification
  • Almost all primes are unramified in a given number field extension

Decomposition of primes

  • Prime ideals in the base field can split, remain inert, or ramify in the extension
  • Splitting behavior described by the factorization of the prime ideal in the extension
  • Decomposition field: largest subfield where the prime splits completely
  • Inertia field: largest subfield where the prime remains unramified

Inertia groups

  • Subgroup of the Galois group that fixes a prime ideal in the extension
  • Measures the ramification behavior of the prime
  • Trivial inertia group indicates unramified prime
  • Size of inertia group related to the ramification index

Zeta functions of number fields

  • Zeta functions are powerful tools for studying arithmetic properties of number fields
  • These functions encode deep information about prime ideals and class numbers
  • Understanding zeta functions is crucial for exploring connections between arithmetic geometry and analytic number theory

Dedekind zeta function

  • Generalization of the Riemann zeta function to arbitrary number fields
  • Defined as a sum over all non-zero ideals in the ring of integers
  • Euler product representation relates zeta function to prime ideals
  • Analytic continuation to the entire complex plane (except for a pole at s=1)

Functional equation

  • Relates values of the zeta function at s and 1-s
  • Involves the discriminant of the number field and gamma factors
  • Symmetry of the provides insights into zeros of the zeta function
  • Crucial for understanding the distribution of prime ideals

Analytic class number formula

  • Expresses the class number in terms of special values of the
  • Involves the regulator and other arithmetic invariants of the number field
  • Generalizes Dirichlet's for quadratic fields
  • Provides a deep connection between algebraic and analytic properties of number fields

Applications in arithmetic geometry

  • Algebraic number fields play a crucial role in various aspects of arithmetic geometry
  • These applications connect number theory with the study of algebraic varieties
  • Understanding these connections is essential for exploring deep problems in mathematics

Elliptic curves over number fields

  • Elliptic curves defined over algebraic number fields instead of Q\mathbb{Q}
  • Points on these curves form finitely generated abelian groups
  • Torsion subgroup structure described by the
  • L-functions of elliptic curves over number fields encode important arithmetic information

Mordell-Weil theorem

  • States that the group of rational points on an elliptic curve over a number field is finitely generated
  • Generalizes the rational case to arbitrary algebraic number fields
  • Rank of the Mordell-Weil group is an important invariant
  • Proof involves descent theory and the weak Mordell-Weil theorem

Height functions

  • Measure the arithmetic complexity of points on algebraic varieties
  • Néron-Tate height on elliptic curves used in the study of rational points
  • Height bounds crucial for effective versions of the Mordell-Weil theorem
  • Generalizations to higher-dimensional varieties (Weil height machine)

Computational aspects

  • Computational techniques are essential for studying algebraic number fields in practice
  • These methods allow for concrete calculations and verifications of theoretical results
  • Understanding computational aspects is crucial for applying number field theory in arithmetic geometry

Algorithms for number field arithmetic

  • LLL algorithm for computing integral bases and solving ideal problems
  • Polynomial factorization algorithms over number fields
  • Methods for computing class groups and unit groups
  • Algorithms for solving norm equations and finding integral points on varieties

Software packages for calculations

  • PARI/GP: specialized system for number theory computations
  • SageMath: comprehensive open-source mathematics software with strong number theory support
  • Magma: commercial system with advanced algorithms for algebraic number theory
  • KANT/KASH: focused on algebraic number theory and function fields

Complexity considerations

  • Many fundamental problems in algebraic number theory are NP-hard
  • Subexponential algorithms exist for some tasks (factoring, class group computation)
  • Quantum algorithms offer potential speedups for certain number-theoretic problems
  • Trade-offs between exact and approximate computations in practice

Key Terms to Review (29)

Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients, making them a key component in the study of number theory and algebraic number fields. These integers generalize the notion of regular integers and play a significant role in various mathematical structures, including unique factorization in Dedekind domains, properties related to units, and connections to zeta functions.
Algebraic Number Fields: Algebraic number fields are extensions of the field of rational numbers, formed by adjoining roots of polynomial equations with rational coefficients. These fields provide a framework for studying the properties of numbers that arise from solving polynomial equations, leading to insights in areas like number theory and algebraic geometry. They are essential for understanding rings of integers, which are integral closures of these fields.
Analytic class number formula: The analytic class number formula is a mathematical relationship that connects the class number of an algebraic number field to its L-functions, particularly the Dedekind zeta function. This formula establishes a link between algebraic number theory and analytic methods, demonstrating how the structure of the field relates to properties of special functions. It reveals the interplay between the arithmetic properties of rings of integers and the analysis of complex functions, playing a crucial role in understanding the distribution of prime ideals in these fields.
Class number: The class number is a fundamental invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It provides crucial insight into the structure of ideal classes within the number field, linking properties of integers to algebraic objects and their behavior under various arithmetic operations.
Class Number Formula: The class number formula relates the class number of a number field to its L-functions and regulators, serving as a critical bridge between algebraic number theory and analytic properties of L-functions. This formula provides insight into the distribution of ideals in a number field and connects the arithmetic of the field with its geometric properties, such as those found in elliptic curves. By analyzing the relationships between class groups and L-functions, this formula highlights the interplay between algebraic structures and their analytical counterparts.
Cyclotomic Field: A cyclotomic field is a special type of number field obtained by adjoining a primitive root of unity to the rational numbers. These fields are generated by the roots of the polynomial equation $x^n - 1 = 0$, where $n$ is a positive integer, and they play a key role in number theory, particularly in understanding the properties of integers and their divisibility.
David Hilbert: David Hilbert was a prominent German mathematician who made significant contributions to various areas of mathematics, including algebraic number theory and algebraic geometry. His work laid the groundwork for many modern theories and techniques, influencing fields such as arithmetic geometry and number theory, which explore the relationships between algebraic structures and their geometric interpretations.
Dedekind Domain: A Dedekind domain is a type of integral domain that satisfies certain properties, including being Noetherian, integrally closed, and having the property that every nonzero prime ideal is maximal. These characteristics make Dedekind domains essential in algebraic number theory, particularly when studying rings of integers and their ideal class groups.
Dedekind zeta function: The Dedekind zeta function is a powerful tool in number theory associated with a number field, defined as a Dirichlet series that encodes information about the field's arithmetic. It generalizes the Riemann zeta function to algebraic number fields, and its properties are closely tied to the structure of the ring of integers within that field, which helps in studying their ideals and class groups.
Degree of an Extension: The degree of an extension refers to the dimension of a field extension as a vector space over its base field. This concept is crucial in algebraic number theory, as it helps to understand how different fields relate to one another and provides insight into the algebraic structure of the numbers involved.
Diophantine equations: Diophantine equations are polynomial equations where the solutions are required to be integers or whole numbers. They are central to number theory and often relate to the search for rational points on algebraic varieties, connecting various mathematical concepts like algebraic geometry, arithmetic, and modular forms.
Dirichlet's Theorem: Dirichlet's Theorem states that there are infinitely many prime numbers in any arithmetic progression of the form $$a + nd$$, where $$a$$ and $$d$$ are coprime integers, and $$n$$ is a non-negative integer. This theorem highlights the distribution of prime numbers and has deep implications in number theory, especially in relation to algebraic number fields and Diophantine approximation.
Discriminant: The discriminant is a mathematical expression that determines the nature of the roots of a polynomial equation, particularly in quadratic forms. It plays a crucial role in various areas of mathematics, helping to identify the types of solutions and their behavior over different fields, especially in algebraic geometry and number theory.
Emil Artin: Emil Artin was a renowned Austrian mathematician known for his contributions to algebraic number theory and class field theory. His work laid the foundation for understanding how number fields behave under various algebraic operations, particularly in relation to the concepts of reciprocity laws and extensions of fields, which are essential in the broader context of rings of integers and algebraic number fields.
Finite extension: A finite extension is a type of field extension where the larger field has a finite-dimensional vector space over the smaller field. This means that the extension can be generated by a finite number of elements, and it plays a critical role in understanding algebraic number theory, particularly when dealing with algebraic number fields, their rings of integers, and local fields. Finite extensions help to analyze how these different structures relate to each other, providing insights into their arithmetic properties and behaviors.
Functional equation: A functional equation is an equation that establishes a relationship between functions and their values at certain points. In the context of various mathematical fields, these equations often reveal deep properties about the functions involved, such as symmetries and transformations, which can be crucial for understanding concepts like zeta functions and L-functions.
Galois Theory: Galois Theory is a branch of mathematics that studies the connections between field theory and group theory, particularly focusing on the symmetries of the roots of polynomial equations. It provides a framework for understanding how different fields are related through their automorphisms and offers insight into the solvability of polynomials by radicals. This theory plays a crucial role in various areas such as algebraic number theory, number fields, and arithmetic geometry.
Height Functions: Height functions are mathematical tools used to measure the complexity of algebraic numbers and points on varieties. They help quantify how 'large' or 'complicated' these numbers or points are, often in relation to their coordinates. This concept is especially useful in number theory and geometry, where understanding the properties of points on elliptic curves, complex tori, arithmetic surfaces, and dynamical systems is crucial for deeper insights into their structure and relationships.
Ideal Class Group: The ideal class group is a fundamental concept in algebraic number theory that measures the failure of unique factorization in a ring of integers of a number field. It consists of equivalence classes of fractional ideals, where two fractional ideals are considered equivalent if their quotient is an invertible ideal. The structure of the ideal class group provides deep insights into the arithmetic properties of the number field, linking to other important concepts like units and their groups, Dedekind domains, and class field theory.
Integral Basis: An integral basis is a set of elements in a ring of integers of an algebraic number field that can be used to express every element of that ring as a unique linear combination of these basis elements with integer coefficients. This concept is crucial because it provides a way to understand the structure and properties of the ring of integers in number fields, helping to generalize results from classical number theory.
Integral Closure: Integral closure is the set of elements in a ring that are integral over that ring, meaning they satisfy a monic polynomial with coefficients from the ring. This concept is crucial in understanding how rings behave within algebraic number fields and how they relate to the rings of integers. Integral closure helps identify extensions of rings that maintain certain algebraic properties, which is particularly important when reducing modulo prime elements.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over a number field is finitely generated. This fundamental result connects the theory of elliptic curves with algebraic number theory, revealing the structure of rational solutions and their relationship to torsion points and complex multiplication.
Norm of elements: The norm of an element in an algebraic number field is a function that assigns a non-negative integer to the element, reflecting the product of its conjugates. It provides important insights into the structure of the number field, revealing relationships between elements and their behavior under multiplication. This concept is crucial for understanding properties like divisibility and prime factorization within algebraic number theory.
Quadratic Field: A quadratic field is a type of number field formed by adjoining the square root of a rational number to the field of rational numbers. This field can be expressed in the form $$ ext{Q}(\sqrt{d})$$, where $$d$$ is a square-free integer. Quadratic fields are significant in algebraic number theory, particularly in understanding the structure of number fields and their rings of integers.
Ramification Theory: Ramification theory studies how prime ideals in a number field behave when extended to a larger field, focusing on the notion of how 'ramified' these primes become. It is crucial for understanding the splitting of prime ideals, the structure of local fields, and how integer rings can be analyzed within algebraic number theory. This theory provides insights into the behavior of extensions and their interaction with different types of rings.
Ring of Integers: The ring of integers in algebraic number theory refers to the set of algebraic integers within a number field, which forms a subring of that field. This concept is crucial for understanding the structure of number fields, as it generalizes the idea of whole numbers to more complex numbers that arise from solutions to polynomial equations with integer coefficients. The properties of this ring are foundational for exploring units and applying Dirichlet's unit theorem.
Separability: Separability refers to a property of algebraic structures, particularly in the context of fields and polynomials, where a polynomial is said to be separable if it does not have repeated roots. This concept is crucial when discussing the nature of extensions in algebraic number fields and the behavior of elliptic curves under isogenies, as it helps to characterize the relationships and morphisms between different algebraic structures.
Trace of Elements: The trace of elements in the context of algebraic number fields refers to the sum of the embeddings of an algebraic element into a number field's completion. This concept captures key information about the roots of a polynomial and helps in understanding the structure of the field itself. The trace is crucial in various calculations, such as determining norms and understanding the behavior of field extensions.
Transcendental Extension: A transcendental extension is an extension of a field that includes elements which are not roots of any non-zero polynomial equation with coefficients in the base field. This concept highlights the distinction between algebraic extensions, where new elements can be expressed as solutions to polynomial equations, and transcendental extensions, where such relationships do not exist. Understanding this distinction is essential when studying the structures of fields and their properties.
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