Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra → Rings and Fields ↓


Field Extensions


In the field of abstract algebra, the concept of "field extensions" is a fascinating and broad topic that explores the depths of mathematical structures beyond the elementary understanding of numbers and operations. Field extensions provide a framework for understanding how larger and more complex fields are built from smaller ones, just as a larger world is built from the microscopic ones. This exposition will discuss in depth what field extensions are, how they differ from vector spaces, and how polynomials play a key role in their construction and understanding.

Field basics

Before we begin exploring field extensions, it is important to understand what fields are. In abstract algebra, a field is a set equipped with two operations: addition and multiplication, for which the set is closed, associative, commutative, has identity elements, and every non-zero element has an inverse.

Examples of regions include:

  • The field of rational numbers ( mathbb{Q} )
  • The field of real numbers ( mathbb{R} )
  • The field of complex numbers ( mathbb{C} )

Defining field extensions

A field extension is a pair of fields ( E supset F ), where the smaller field ( F ) is a subfield of the larger field ( E ). This means that the operations and elements of ( F ) are contained in ( E ), and the operations of ( F ) are the same within ( E ). The field ( E ) is called an extension of ( F ).

Consider the simple example of the field of rational numbers: ( mathbb{Q} ).

[ mathbb{Q} = { a/b | a, b in mathbb{Z}, b neq 0 } ]

The field extension may be the field of real numbers, containing all rational numbers and additional elements that complete it as a set of real numbers.

Visualizing field extensions

To understand the concept of field extension, imagine a circle (denoted as (mathbb{Q}) ) within a larger circle (denoted as (mathbb{R}) ). Everything that is in the smaller circle of (mathbb{Q}) ) is part of the larger circle of (mathbb{R}) ), but there are other elements in (mathbb{R}).

(mathbb{Q}) (mathbb{R})

Degree of field expansion

The degree of a field extension refers to the dimension of the larger field (E) as a vector space over the smaller field (F). If (E) is a finite-dimensional vector space over (F), then the extension is said to have finite degree.

The degree is denoted as ([E:F]). For example, if ( E = mathbb{Q}(sqrt{2})), the field obtained by adjoining (sqrt{2}) to (mathbb{Q}), has degree ([E:F] = 2) since the base is ({1, sqrt{2}}).

Construction of field extensions

After understanding what fields and field extensions are, we move on to constructing field extensions. The process of summation involves adding elements to the field so that it becomes larger and possibly contains adjacent roots of polynomials.

For example, consider the polynomial:

f(x) = x^2 - 2

This polynomial has no roots in (mathbb{Q}), but we can "extend" (mathbb{Q}) to include the roots. The field extension (mathbb{Q}(sqrt{2})) contains all numbers of the form (a + bsqrt{2}) where (a, b in mathbb{Q}).

Visual example of simple field extensions

Imagine a number line for (mathbb{Q}) and an extended number line consisting of points like (sqrt{2}).

0 1 -1 (sqrt{2})

Algebraic versus transcendental extension

Field extensions can be classified into two types: algebraic and transcendental extensions. An element (alpha) in a field extension (E) of a field (F) is called algebraic over (F) if it satisfies a polynomial equation with coefficients in (F).

Consider the example of (mathbb{Q}(sqrt{2})) where (sqrt{2}) is algebraic over (mathbb{Q}) because it satisfies:

[sqrt{2}^2 - 2 = 0]

However, in the extension (mathbb{Q}(pi)), (pi) is not algebraic over (mathbb{Q}) and is hence called a transcendental number because there is no polynomial with rational coefficients for which (pi) is a root.

Text example of an extended field

Let us explore field expansion using the example of square roots:

Example: Find the field extension ( F = mathbb{Q}(sqrt{5}) ).

Solution: Start with (mathbb{Q}), which is the rational numbers. The extension field (mathbb{Q}(sqrt{5})) contains all elements of the form (a + bsqrt{5}), where (a, b in mathbb{Q}).

This is because a normal element of this field can be obtained by taking rational linear combinations of the basis elements ({1, sqrt{5}}). Thus, ([F:mathbb{Q}] = 2).

Role of polynomials in area expansion

Polynomials are central to understanding and working with field extensions. An essential concept is the minimal polynomial of an element (alpha) within an extension field (E) over a field (F).

The minimal polynomial of an element (alpha) is the monic polynomial of lowest degree in (F[x]) whose (alpha) is a root.

For example, if (alpha = sqrt{5}), then the minimal polynomial on (mathbb{Q}) is:

x^2 - 5

Finding these minimal polynomials helps in formulating and understanding field extensions because they provide the constraints needed to solve and express elements over a larger field.

Summary and closing

Field extensions are a powerful concept in abstract algebra, providing a way to systematically extend the scope of fields to include more numbers while respecting the underlying structures and operations. From basic operations on rational numbers to the inclusion of more exotic numbers such as imaginary or complex numbers, or transcendental numbers, field extensions provide an essential bridge toward reconciling the finite and the infinite in mathematical thought.

They play a vital role in many areas of mathematics and its applications, including solving polynomial equations, understanding the structure of numbers, and working on more complex problems in algebra, number theory, and beyond. A strong understanding of field theory facilitates further exploration of deeper mathematical concepts and advanced applications.


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Field Extensions

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