PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Number Theory → Algebraic Number Theory ↓


Class Groups


Algebraic number theory is a practical area of mathematics that extends familiar notions of numbers and arithmetic to more abstract and generalized settings. One of the central concepts in this area is the class group. Class groups provide a powerful tool for understanding the arithmetic behavior of algebraic numbers, especially with respect to the factorization of integers in the domain of algebraic number fields.

Understanding number fields

To delve deeper into class groups, it is first important to understand what number fields are. A number field is a finite extension of the field of rational numbers (Q). An example of a number field is the quadratic field, which is generated by adding the square root of a number to Q. Formally, if d is not a perfect square, then Q(√d) is a quadratic field.

Example: Q(√2) is the set of all numbers of the form a + b√2, where a and b are rational numbers.

The number field K has an associated ring of integers, denoted by OK. This ring generalizes the ordinary integers Z within a number field and consists of elements that are roots of monic polynomials (polynomials with leading coefficient 1) with coefficients in Z.

Introduction to ideals

The cornerstone of understanding class groups is the concept of ideals in the ring of integers OK. An ideal is a subgroup of OK that satisfies certain algebraic properties. In particular, ideals allow us to discuss divisibility and factorization in a more generalized way.

For example, consider the set of multiples of a single element of OK, say α. This is an ideal, sometimes written as αOK. Unlike elements, which cannot have unique factorizations in OK, ideals can always be uniquely factorized into prime ideals, which is an important advantage.

OK: the ring of integers I: Ideal in O.K. I1, I2 ...: prime ideals

Prime ideals and factorization

In the ring of integers OK, unlike the ring of integers Z, every element cannot be uniquely partitioned into prime elements. This is where the concept of ideals becomes essential. Unlike elements, an ideal in OK can always be identified as a unique product of prime ideals.

Example: In Z[√-5], the element 6 can be factored as 2 × 3 and also as (1 + √-5)(1 - √-5).

In higher-degree number fields the situation becomes even more complicated, but the beautiful property of unique factorization of primes is retained, ensuring a well-defined concept of divisibility.

Defining a class group

The class group of a number field K is a measure of how far the ring of integers OK is from the unique factorization property of the integers Z. The class group is defined as the quotient group of the set of fractional ideals by the set of prime ideals.

Class group = Fractional Ideals / Principal Ideals

Here, a partial ideal is a generalization of an ideal that may not lie entirely within OK, but can become an ideal of OK by multiplication by some non-zero element of OK. A prime ideal is simply an ideal generated by an element of OK.

Importance of class number

The size of the class group, known as the class number, is an important invariant of the number field. A class number of 1 means that every ideal in OK is a prime ideal, and therefore, OK enjoys unique factorization.

Example: The field Q(√-1) has a class number of 1, meaning Z[√-1] has unique factorization.

Higher class numbers indicate the existence of non-prime ideals, which implies more complex structure. Understanding this complexity is a major focus of algebraic number theory.

Class group: O measures factorization in K prime ideal ↔ unique factorization Partial ideal ↔ All ideal

Working with class groups

Discovering the properties of class groups is not just a theoretical pursuit; these groups have profound implications in diverse fields such as algebraic geometry and cryptography.

Calculating class numbers

Computing the class number of a number field can be highly non-trivial and involves intensive techniques from both algebra and analysis. Algorithms such as the Minkowski bound and the use of L-functions are used to compute class numbers.

A basic method involves investigating the discriminant of a number field. The discriminant provides some initial estimates that are useful in establishing bounds for the class number.

Example: For the quadratic field Q(√d), the class number can often be shown to be 1 or 2 using discriminants and congruences.

Applications of class groups

Class groups play a role in various deep questions about number fields. They are essential in the proof of Fermat's Last Theorem for certain classes of numbers, and they have interesting applications in coding theory, where structure and symmetry can be important for strong communication.

Beyond theoretical aspects, insights gained from class groups contribute to a broader understanding of cryptographic systems, which often rely on properties of algebraic structures influenced by class groups.

Conclusion

The study of class groups reveals a rich fabric of algebraic structures beyond the integers, providing a glimpse into the diverse world of algebraic number theory. The interrelationship between ideals, factorization, and class numbers is a beautiful demonstration of mathematics' power of generalization and integration. Through class groups, we understand not only the behavior of numbers, but also the underlying symmetries and complexities of the systems they belong to.


PHD → 5.3.3


U
username
0%
completed in PHD

← Prev (5.3.2)
Ideals
Next (5.3.4) →
Galois Representations

Comments 



Class Groups

https://www.buddymath.com/phd/5.3.3?bmal=en