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 → Algebra → Field Theory ↓


Field Extensions


In mathematics, particularly in abstract algebra, fields are powerful structures that allow us to understand a variety of mathematical concepts. A field can be thought of as a set equipped with two operations, addition and multiplication, that satisfy certain properties. An example of a field is the set of rational numbers, denoted by ℚ. Now, imagine how we might explore larger fields that contain additional elements not present in our original field. This exploration leads us to the concept of field extension.

A field extension is essentially a larger field containing a smaller field. If we denote the smaller field by F and the larger field by K, we say that K is an extension of F, written as K/F. The study of field extensions provides insight into the structure and behavior of fields, leading to profound results in algebra.

Definitions and basic concepts

To understand field extension, let's start with some definitions:

Field: A field is a set F having two operations, addition and multiplication, that satisfy the following properties:

  • Closure under addition and multiplication
  • Associativity of addition and multiplication
  • Commutativity of addition and multiplication
  • Existence of additive and multiplicative identities
  • Existence of additive inverse for every element
  • Existence of multiplicative inverse for every nonzero element
  • Distributive Law: a(b + c) = ab + ac

Field extension: Given a field F, a field K is called an extension of F if K contains F and has the structure of a field. In other words, F is a subfield of K

The degree of an extension, denoted by [K : F], is the dimension of K as a vector space over F. If the extension is finite, we can write: [K : F] = n, where n is a positive integer. An infinite extension has no finite degree.

Examples of field extensions

Example 1: Complex numbers over the real numbers

One of the most famous examples of a field extension is the field of complex numbers, denoted by ℂ over the field of real numbers ℝ. Complex numbers include numbers of the form a + bi where a and b are real numbers, and i is an imaginary unit with the property i 2 = -1.

ℂ = ℝ(i)

Here, the degree of the extension [ℂ : ℝ] is 2 since any complex number can be expressed as a linear combination of 1 and i on ℝ:

a + bi = a*1 + b*i

This expression shows that {1, i} is a basis for ℂ on ℝ.

Example 2: Algebraic expansion

Consider the field of rational numbers ℚ and the polynomial x 2 - 2. This polynomial does not have roots in ℚ. To find a field extension that contains the roots of this polynomial, we can consider the field ℚ(√2), which consists of numbers of the form a + b√2 where a and b are rational numbers.

ℚ(√2) = { a + b√2 | a, b in ℚ }

In this case, [ℚ(√2) : ℚ] = 2 since √2 is a root of the polynomial x 2 - 2, and every element of ℚ(√2) can be uniquely expressed as a linear combination of 1 and √2 on ℚ.

ℚ ℚ(√2)

Types of field extensions

Finite extension

An extension K/F is called finite if the degree [K : F] is finite. The applications of finite extensions are in coding theory, cryptography, and more.

Infinite extensions

An extension K/F is called infinite if [K : F] is infinite. Infinite extensions are important in the study of transcendence and algebraic freedom.

Algebraic extension

An element α in K is algebraic over F if it is a root of a non-zero polynomial with coefficients in F. An extension K/F is called algebraic if every element of K is algebraic over F

Transcendental extension

An extension K/F is transcendental if there exists at least one element in K that is not algebraic over F. This means that the element cannot be a root of any polynomial with coefficients in F

Properties of field extensions

Tower law

The Tower Law is an important property of field extensions that relates the degree of the extension. If we have extensions L/K and K/F, then the Tower Law states:

[L : F] = [L : K] * [K : F]

This means that the total degree of the extension is the product of the individual degrees.

Simple extensions

A simple extension is a field extension that can be generated by adding a single element to the base field. For example, the field ℚ(√2) is a simple extension of ℚ generated by √2.

F(α) = { f(α)/g(α) | f, g are polynomials in F }

Here, α is a root of some polynomial in F, and the above notation denotes all elements that can be expressed as ratios of polynomials with coefficients in F

F f(α) K

Applications of field extension

Field extensions have many applications in various branches of mathematics and related fields:

Galois theory

Field extensions form the basis for Galois theory, which explores the connections between field extensions and group theory. Galois theory helps in understanding the solvability of polynomial equations and in finding the roots of polynomials.

Solving polynomial equations

Field extensions are fundamental in solving polynomials that cannot be solved within the boundaries of the original field. For example, the extension enables one to solve ℚ(√2) x 2 - 2 = 0.

Algebraic number theory

Field extensions play an important role in algebraic number theory, where integers are generalized to the algebraic integers, and solutions of equations are investigated over large fields.

Conclusion

Field extensions are the cornerstone of field theory, providing essential tools for enhancing our understanding of algebraic structures. By exploring field extensions, we uncover complex relationships between different fields and gain deep insights into polynomial equations, algebraic numbers, and much more. The study of field extensions remains a vibrant and essential area in pure mathematics, influencing a variety of applications in science and engineering.


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Field Extensions

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