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 ↓


Algebraic Closures


Algebra is a vast area of mathematics that deals with numbers, equations, and the rules for manipulating these things. In the branch of algebra called "field theory," we study fields, which are sets we can add, subtract, multiply, and divide (except by zero).

An important question in field theory is about solving polynomial equations. For example, we might ask:

x^2 - 2 = 0

To solve this, we need to find numbers that make the equation true. A field is called algebraically closed if every non-constant polynomial with coefficients in that field has a root in the same field. This raises a general mathematical question: given a field, can we find a larger field, called its algebraic closure, that has a root for every polynomial?

Basic concepts

To understand algebraic closure in more depth we need to understand some key terms and concepts.

  • Field: A set with two operations, addition and multiplication, that satisfy certain rules such as distributivity, commutativity, associativity, existence of identity elements, and inverses.
  • Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
  • Root of a polynomial: The value for which the polynomial evaluates to zero.

Now with an understanding of these terms, we can consider algebraic closure.

Examples of fields and extensions

The simplest field to consider is ℚ (the set of all rational numbers). This is an example of a field, but it is not algebraically closed. Consider the equation:

x^2 - 2 = 0

The solutions of this equation are √2 and -√2, neither of which are rational numbers. Thus, ℚ is not a solution of this polynomial equation, which means that it is not algebraically closed.

Similarly, the field of real numbers ℝ is also not algebraically closed. To find an example of failure, consider:

x^2 + 1 = 0

Its solutions are i and -i. These solutions are not real numbers, so the real numbers also do not have every root for every polynomial.

Algebraic closure and complex numbers

The smallest field that contains all the other fields we have discussed and that is algebraically closed is the field of complex numbers, denoted by ℂ.

Every polynomial equation with complex coefficients has a root in ℂ. Geometrically, if we plot complex numbers, the complex field ℂ can be thought of as a plane with the x-axis representing the real part and the y-axis representing the imaginary part.

Real part Imaginary part 1 + i

The red dot on the plane represents the complex number 1 + i. Any polynomial like x^2 + 1 has roots in the complex number field i and -i, which shows that the complex numbers form an algebraic closure for ℝ.

Construction of the algebraic closure

Constructing the algebraic closure of a given field involves extending the field to include roots for those polynomials that do not already have roots within the field. In simple terms the construction process is as follows:

  • Start with your base field, say the rational numbers ℚ or the real numbers ℝ.
  • Consider all polynomial equations, even those whose roots do not lie in the existing field.
  • Add the roots of the new polynomials one by one to form successive field expansions.
  • Continue this iterative process until no new roots are needed.

This idea just scratches the surface of the vast and deep field of algebraic methods, where certain areas are combined over and over again, and new areas involve roots of more and more polynomials.

Applications of algebraic closure

Algebraic closures have many practical applications in advanced mathematics and even physics. Some common applications include:

1. Solving polynomial equations

As mentioned earlier, the main purpose of considering algebraic closure is to ensure that every polynomial equation has a solution. This idea is fundamental in algebra and calculus.

2. Galois theory

Galois theory is a notable topic in algebra that investigates how algebraic endowments help in understanding the solvability of polynomial equations. This theory uses field extensions including algebraic endowments to reveal deep insights into the symmetry of the roots of polynomials.

3. Complex analysis

Complex analysis often treats complex numbers as inherently vivid and important, in particular, in understanding functions of complex numbers. Algebraic closure is important in discussions involving functions defined by power series or polynomial equations.

4. Algebraic geometry

Algebraic geometry studies solutions of polynomials and equations, often involving multivariate systems. The completeness property of algebraic endowments is important when considering systems over fields.

Conclusion

Understanding algebraic closures and their consequences is crucial in exploring the vast expanses of advanced mathematics. Whether working through algebraic geometry, complex analysis, or field theory, the guarantees of solutions provided by algebraic closures stand as a foundation for developing further in-depth mathematics.

The concept of algebraic closure not only binds fields into coherent unified structures, but also sheds light on the systematic progress of solving polynomial equations, which is one of the many beautiful aspects of mathematical theory.


PHD → 1.3.4


U
username
0%
completed in PHD

← Prev (1.3.3)
Galois Theory
Next (1.3.5) →
Finite Fields

Comments 



Algebraic Closures

https://www.buddymath.com/phd/1.3.4?bmal=en