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 ↓


Number Fields


In algebraic number theory, a number field is a concept that extends the idea of rational numbers. Mathematical development has gone beyond integers and rational numbers to include fields made up of algebraic numbers. It is a broad, rich area of mathematics that has attracted attention because of its beauty and application to solving complex problems such as Fermat's Last Theorem.

To really understand what a number field is, let us start by considering the rational numbers, denoted as Q. Rational numbers are numbers that can be expressed as a fraction a/b where a and b are integers and b is not zero. These numbers form a field because you can add, subtract, multiply, and divide any two rational numbers (except by zero) and always get another rational number.

Basic definitions

In mathematics, a field is a set equipped with two operations, addition and multiplication, that satisfy certain properties: commutativity, associativity, distributivity, and the existence of additive and multiplicative identities and inverses. The rational numbers are a familiar example of a field.

1. Closure: For all a, b in a field F, both a + b and a * b are in F. 
2. Associativity: For all a, b, c in F, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c). 
3. Commutativity: For all a, b in F, a + b = b + a and a * b = b * a. 
4. Identity elements: There exist distinct elements 0 and 1 in F such that for all a in F, a + 0 = a and a * 1 = a. 
5. Inverses: For all a in F, there exist elements -a and a^(-1) in F such that a + (-a) = 0 and a * a^(-1) = 1, a^(-1) exists if a != 0. 
6. Distributivity: For all a, b, c in F, a * (b + c) = a * b + a * c.

Number fields are extensions of the field of rational numbers. In concrete terms, a number field is formed by taking rational numbers and adjacent roots of polynomials with coefficient Q.

More formal definition of a number field

Formally, a number field is a finite degree field extension of Q. This means that if you have a number field K, then there is some integer n such that K can be viewed as a vector space of dimension n over Q.

For example, consider the field Q(sqrt{2}), which consists of all numbers that can be written in the form a + b sqrt{2}, where a and b are rational numbers. This is a number field of degree 2 over Q, since we can think of Q(sqrt{2}) as a 2-dimensional vector space over Q.

Example: constructing a number field

Let's create a simple number field. Consider the polynomial x^2 - 2 = 0. The solutions of this polynomial are sqrt{2} and -sqrt{2}. By connecting the roots of this polynomial to Q, we get the field Q(sqrt{2}).

The elements of Q(sqrt{2}) take the form a + b sqrt{2}, where a and b are rational numbers. In this context, Q(sqrt{2}) is a degree 2 extension because it is generated by an extension of Q with roots of degree 2 polynomials.

Rational field (Q) The number field Q(sqrt{2})

Properties of number fields

Number fields inherit many properties from fields. Some of the important properties are as follows:

  • Structure: As a field, its elements can be added, subtracted, multiplied, and divided (except by zero), and its structure is based on its degree of extension.
  • Closure: Number fields are closed under their operations, which means that any combination of their elements with field operations will give another element within the number field.
  • Existence of inverses: every non-zero element has a multiplicative inverse within the field.
  • Finiteness: A number field is finite-dimensional when viewed as a vector space over Q.

Primitive elements

In any finite extension of a number field, there is a special element known as the primitive element. This field can be generated purely by adding this one element to Q. If K = Q(α), then α is a primitive element.

For example, let us consider the field Q(sqrt{3}). The element sqrt{3} is a primitive element, and any element in Q(sqrt{3}) can be expressed in terms of sqrt{3} and the rational numbers.

Q(sqrt{3}) Primitive element: sqrt{3}

Galois group

The Galois group of a number field is a concept that describes symmetries in the roots of polynomials. For a number field K that is a normal extension of Q, the Galois group Gal(K/Q) consists of all field automorphisms of K that keep Q constant.

The Galois group provides great insight and powerful tools for studying the field structure and the relationship between different number fields. For example, it can help to understand the solvability of polynomials, among other things.

For example: Polynomial f(x) = x^2 - 2. 
Number Field: Q(sqrt{2}). 
Galois Group: {1, -1}, representing the identity and the negation of square root.

Applications of number fields

Number fields are much more than a theoretical curiosity. They have been important in solving many landmark mathematical problems, including:

  • Fermat's Last Theorem: solved this problem with the insight that the solutions are related to the algebraic integers, which are integral elements in number fields.
  • Class field theory: This is a major area of study that investigates abelian extensions of number fields.
  • Elliptic curves: They form external structures that are defined over number fields and have implications for cryptosystems.

Understanding number fields has inspired developments in computational number theory and algorithmic approaches to solving Diophantine equations. Their study is a highly fertile ground for advancing modern mathematics.

The nature of number fields is closely connected to other areas of mathematics, such as algebraic geometry, where fields of functions on curves are essentially number fields. Thus the study of number fields is connected to many parts of modern number theory.

Conclusion

Number fields are the fundamental basis of algebraic number theory, providing a deeper insight into the nature of algebraic numbers. By expanding the rational numbers through polynomial roots, number fields open the door to a deeper understanding of mathematics and the solution of many classical problems.

The field continues to grow as mathematicians delve further into concepts such as higher-dimensional field extensions, p-adic number fields, and more complex Galois theories. The equations, structures, and theories formulated through the richness of number fields continue to be central to countless mathematical investigations and discoveries.


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