Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra ↓


Abstract Algebra


Abstract algebra is a fascinating field of mathematics that studies algebraic structures beyond the elementary arithmetic and algebra learned in high school. In high school, students deal mostly with the algebra of numbers and functions, while in abstract algebra, we delve into more complex systems like groups, rings, and fields. These structures help us understand and formalize concepts that are the basis of modern mathematics and many practical applications in computer science, physics, and engineering.

Algebraic structures

An algebraic structure consists of a set equipped with one or more operations. The study of these structures is central to abstract algebra. Here are some of the most important algebraic structures:

Group

A group is a set of elements combined by an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility.

Formally, a group is a set G with a binary operation * such that:

1. Closure: For every a, b in G, the result of the operation a * b is also in G
2. Associativity: for every a, b, c in G, (a * b) * c = a * (b * c)
3. Identity: There exists an element e in G such that for every a in G, e * a = a * e = a.
4. Invertibility: for every a in G, there exists an element b in G such that a * b = b * a = e (where e is the identity element).

Example: Consider the group Z of integers under addition. It forms a group because it satisfies all four properties: closure (the sum of any two integers is an integer), associativity, identity (the integer 0, since any integer remains unchanged when 0 is added), and invertibility (for any integer a, there exists -a such that a + (-a) = 0).

Rings

A ring is an algebraic structure consisting of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. A ring must satisfy several properties, including the presence of an additive identity and the distributive property.

Formally, a ring is a set R with two operations: addition + and multiplication *, such that:

1. (R, +) is an abelian group.
2. Multiplication is associative.
3. Distributive laws apply: for all a, b, c in R, a * (b + c) = a * b + a * c and (b + c) * a = b * a + c * a.

Example: The set of all integers Z with the usual addition and multiplication operations is a ring.

Field

A field is an algebraic structure in which addition, subtraction, multiplication, and division are defined and behave in the same way as they normally do with rational and real numbers. A field is a special kind of ring.

Formally, a field is a set F with two operations (addition and multiplication) such that:

1. (F, +) is an abelian group.
2. (F  {0}, *) is also an abelian group.
3. The distribution rules apply as stated in the ring.

Example: The set Q of rational numbers is a field because it allows addition, subtraction, multiplication, and division (except by zero), and satisfies all the properties of a field.

Visual representation

Group Rings Field

The diagram above clearly shows the relationship between groups, rings, and fields. Note how every field is a ring, and every ring includes the concept of a group because of its additivity properties.

Importance in mathematics

Abstract algebra plays a vital role in modern mathematics. It provides the framework for unifying many advanced mathematical concepts. Here are some reasons why abstract algebra is so essential:

  • This leads to a comprehensive understanding of number systems and the arithmetical rules that govern them.
  • Abstract algebra forms the basis for more advanced mathematical theories such as topology, algebraic geometry, and cryptography.
  • Its concepts are used in a variety of applied mathematical fields, including coding theory, signal processing, and the development of mathematical proofs.

Conclusion

Abstract algebra is a rich and deep field of mathematics that goes beyond numbers to study more general algebraic structures such as groups, rings, and fields. By understanding these structures, mathematicians can solve complex problems and develop new mathematical theories that impact technology and science. The journey into abstract algebra is both challenging and rewarding, as it opens up new perspectives in understanding the world of mathematics.


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