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


Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study integers, rational numbers, and their generalizations. It has become a foundational field in advanced mathematics because of its relevance in solving classical problems and its applications in modern fields such as cryptography, coding, and beyond.

Basic concepts and definitions

Before delving deeper into algebraic number theory, let's understand some basic terms and concepts:

Number fields

A number field is a finite degree field extension of the rational numbers ℚ. In simple terms, it is a large set of numbers that includes the rational numbers and is closed under addition, subtraction, multiplication, and division. Number fields are often denoted as K

Algebraic numbers

An algebraic number is a number that is a root of a nonzero polynomial equation with rational coefficients. For example, the solutions of x^2 - 2 = 0 are √2 and -√2, both of which are algebraic numbers.

Ring of integers

Within the number field, an important subset is the ring of integers, denoted ℤ K. This set generalizes ordinary integers in the context of the number field. It is important in understanding the factorization and divisibility properties.

Integers and factorization

Factorization plays an important role in algebraic number theory. Unlike the rational integers, integers in number fields do not always have unique prime factorization. This leads to many interesting problems and structures.

Example: Factorization into Gaussian integers

Consider the Gaussian integers, which are complex numbers of the form a + bi where a, b are integers and i is the imaginary unit. For example, can be uniquely divided in 5 different ways:

5 = (2 + i)(2 – i)
5 = (1 + 2i)(1 – 2i)

Here, the factors are Gaussian integers, and each factorization can be thought of as a "prime" into the Gaussian integers, indicating the lack of a unique factorization.

Sum of two squares

Algebraic number theory also provides tools for solving classical problems such as which integers can be expressed as the sum of two squares. It turns out that an integer n can be expressed as the sum of two squares if for every prime p that divides n and p ≡ 3 (mod 4), the exponent of p in the prime factorization of n is even.

Ideal and unique factorization

To address the issue of unique factorization, algebraic number theory introduces ideals. An ideal is a subset of a ring that generalizes the notion of numbers being "evenly divisible" by other numbers.

Prime ideal

An ideal is prime if the product of two elements that are in the ideal has at least one of those elements in the ideal. Unique factorization of ideals is maintained, unlike elements of a ring, which provides a powerful tool for analysis.

Class groups and units

Class groups and unit theory form the basis of the study of the arithmetic of number fields.

Class group

The class group is an algebraic structure that measures the failure of unique factorization in the ring of integers of a number field. It classifies all ideals of the ring into classes of equivalent ideals.

Units

The units of a ring are its elements that have multiplicative inverses. Describing the structure of the unit group of the ring of integers of a number field is a central problem in algebraic number theory.

Application

There are profound applications of algebraic number theory:

  • Cryptography: The RSA encryption system and elliptic curve cryptography are based on algebraic number theory.
  • Coding principles: Helps in error detection and correction in data transmission.
  • Algebraic geometry: connections with algebraic curves and varieties.

Conclusion

Algebraic number theory provides a fascinating glimpse into the deeper structures underlying even the simplest numerical systems. As abstract as it may seem, its real-world applications and the powerful insights it provides make it a vibrant area of mathematical research.


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Algebraic Number Theory

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