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 ↓


Galois Theory


Galois theory, named after the mathematician Évariste Galois, is a rich area of abstract algebra that studies the symmetries of polynomial equations. It provides a deep connection between field theory and group theory, giving insight into the solvability of equations by radicals.

Introduction to fields and field extensions

A field is a set equipped with two operations, addition and multiplication, that satisfy certain axioms similar to those of the familiar arithmetic of rational numbers. The simplest example of a field is the set ℚ of rational numbers. Other examples include the field ℝ of real numbers, the field ℂ of complex numbers, and finite fields.

Field example: ℚ

A field extension is a larger field that contains a smaller field as a subfield. For example, ℝ is a field extension of ℚ because it contains all rational numbers as well as additional elements such as √2.

Field extension: ℝ ⊃ ℚ

Polynomial equations and roots

Consider a polynomial equation such as x 3 - 2 = 0 The roots of this polynomial are the solutions of this equation. In the field of rational numbers ℚ, this polynomial has no roots, since the cube of any rational number does not equal 2. However, if we extend our field to include the complex numbers, we can find solutions, such as √[3]{2}.

Solvable by radicals

A polynomial equation is called solvable by radicals if its roots can be expressed using a finite combination of addition, subtraction, multiplication, division, and taking nth roots. For example, the quadratic polynomial x 2 - 4 = 0 is solvable by radicals, since its solutions are ±2.

Solvable by radicals: x 2 -4=0 ⇒ x=±2

Galois group

The main idea of Galois theory is to study the symmetries of the roots of a polynomial equation through the concept of the Galois group. The Galois group is a group of permutations of the roots of a polynomial that preserves the algebraic relations between these roots.

To construct the Galois group for a polynomial, consider the roots of the polynomial as elements of a larger field known as the splitting field. Each element (permutation) of the Galois group corresponds to an automorphism of this field that fixes the base field.

Visualization of Galois groups

Let's look at this concept with a simple polynomial x 2 - 2 = 0 Its roots are ±√2. The splitting field for this polynomial on ℚ is ℚ(√2).

Root: ±√2, Division area: ℚ(√2)

The Galois group here has two automorphisms: the identity (leaving each element unchanged) and another that replaces √2 by -√2. This group is isomorphic to the cyclic group of order 2, denoted by 

C 2
.

Fundamental theorem of Galois theory

One of the main results of Galois theory is the fundamental theorem of Galois theory. It states that there is a one-to-one correspondence between subfields of a field extension and subgroups of the associated Galois group.

This correspondence is inclusion-reversal, meaning that a smaller subgroup corresponds to a larger subfield, and vice versa. This powerful theorem helps us understand complex field extensions by studying simple group structures.

Example: Solving the quinary equation

Historically, the best-known application of Galois theory is solving polynomial equations of high degree. For example, Galois theory provides insight into the question of whether the general quintic equation can be solved by radicals.

A general quinary equation is of the form:

 ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0
Galois theory shows that for a general polynomial of degree five (or higher), there is no formula, similar to the quadratic formula, that uses only basic arithmetic and root extraction to solve it for all cases. This is because the Galois group of a general quintet is the symmetric group S 5, which is not a solvable group.

Example of Galois extensions

Suppose we want to understand the extension of ℚ ℚ(√3). The roots of the polynomial x 2 - 3 = 0 are ±√3. The Galois group for this extension is also C 2, which is the same as the example above, since the same reasoning applies: the isomorphism involves swapping the roots.

Applications and implications

Galois theory has a profound impact not only in pure mathematics but also in fields such as cryptography and coding theory. It helps in understanding the structure of field expansions, solving polynomial equations, and analyzing symmetries in various mathematical structures.

In cryptography, finite fields, which are central to many cryptographic systems, can be analyzed using the tools of Galois theory to ensure security and efficiency. In coding theory, an understanding of polynomial roots can enhance error detection and correction methods.

Concluding remarks

Galois theory is a deep and sophisticated area of algebra that combines field theory and group theory. It explains the nature of polynomial equations, providing a powerful framework for understanding the solvability of equations and the structure of field extensions.

By studying the symmetries of equations through Galois groups, mathematicians can gain deep insights into the universe of numbers and discover beautiful solutions to complex algebraic problems.


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