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 ↓


Fields


In the world of mathematics, abstract algebra is a powerful tool that helps us delve deeper into the fundamental structures that shape algebraic systems. A key component of abstract algebra is the concept of a "field." A field is a fascinating structure that extends our understanding of the number systems we use regularly, such as integers, rational numbers, real numbers, and complex numbers.

Basic definition of the area

A field is a set equipped with two binary operations, commonly referred to as addition and multiplication, that satisfy specific properties. These properties are:

  1. Closure: For any two elements (a) and (b) in a field (F):
    • Their sum (a + b) is also in (F).
    • Their product (a cdot b) is also in (F).
  2. Associative property:
    • For addition: ((a + b) + c = a + (b + c))
    • For multiplication: ((a cdot b) cdot c = a cdot (b cdot c))
  3. Commutative property:
    • For addition: (a + b = b + a)
    • For multiplication: (a cdot b = b cdot a)
  4. Identity element: There are two special elements in (F):
    • Additive identity: There exists an element 0 such that for all (a in F), (a + 0 = a).
    • Multiplicative identity: There exists an element 1 such that for all (a in F), (a cdot 1 = a). Note that (0 neq 1).
  5. Inverse elements: For every element of the field:
    • Additive inverse: For every (a in F), there exists an element (-a) such that (a + (-a) = 0).
    • Multiplicative inverse: For every (a neq 0) in (F), there exists an element (a^{-1}) such that (a cdot a^{-1} = 1).
  6. Distributive property: Multiplication distributes over addition: 
                A cdot (B + C) = A cdot B + A cdot C

Examples of fields

Rational numbers ((mathbb{Q}))

The set of all rational numbers, denoted by (mathbb{Q}), forms a field. Rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero.

Let's look at field properties using rational numbers:

  • Conclusion: The sum or product of any two rational numbers is also a rational number.
  • Associative, commutative and distributive properties: These are well-known properties of addition and multiplication applied to rational numbers.
  • Identity element: The additive identity is 0, and the multiplicative identity is 1.
  • Inverse element: Every rational number has an additive inverse (that is, (-a)) and every nonzero rational number has a multiplicative inverse (that is, if (a neq 0), then (a^{-1}) exists).

Real numbers ((mathbb{R}))

The set of all real numbers, denoted by (mathbb{R}), is another example of a field. The real numbers include all rational numbers and all irrational numbers (numbers that cannot be expressed as simple fractions).

The properties of a field apply to real numbers just as they do to rational numbers. You can verify each of these properties by testing different real numbers.

Complex numbers ((mathbb{C}))

The set of all complex numbers, denoted by (mathbb{C}), also forms a field. A complex number is defined as (a + bi), where (a) and (b) are real numbers, and (i) is an imaginary unit having the property (i^2 = -1).

    (a + b) + (c + d) = (a + c) + (b + d)i
    (a + bi) cdot (c + di) = (ac - bd) + (ad + bc)i

As in other fields, you can investigate closure, associative, commutative, identity element, inverse element, and distributive rules for complex numbers.

Non-instances of fields

Integer ((mathbb{Z}))

Although it might seem that the integers could form a field, this is not the case. This is because not every non-zero integer has a multiplicative inverse within the integers.

For example, consider the integer 2. There is no integer (x) such that satisfies the following:

    2 cdot x = 1

Solution (x = 0.5) is not an integer.

Visual example of fields

The field can be understood better by visualizing the operation. Below is a simple visualization for addition and multiplication in the field of real numbers.

A B A+B A B A*B

In this visualization, the blue, red, and green dots represent numbers on the real number line. The top number line represents the addition operation, where numbers are added together to form a third number in the field. The bottom number line represents multiplication within the real numbers.

Finite fields

A finite field, also known as a Galois field, is a field that has a finite number of elements. The most common example of a finite field is the group of integers (mathbb{Z}_p) that are under addition and multiplication modulo a prime number (p).

Example: (mathbb{Z}_5)

The set (mathbb{Z}_5) contains the numbers {0, 1, 2, 3, 4}, which are added and multiplied at a modulus of 5. Below, we show the addition and multiplication tables for (mathbb{Z}_5).

    Totals table:
      + | 0 1 2 3 4
      ,
      0 | 0 1 2 3 4
      1 | 1 2 3 4 0
      2 | 2 3 4 0 1
      3 | 3 4 0 1 2
      4 | 4 0 1 2 3

    Multiplication table:
      × | 0 1 2 3 4
      ,
      0 | 0 0 0 0 0
      1 | 0 1 2 3 4
      2 | 0 2 4 1 3
      3 | 0 3 1 4 2
      4 | 0 4 3 2 1

In (mathbb{Z}_5), every non-zero element has a multiplicative inverse, satisfying all field properties. For example, the inverse of 3 is 2 because:

    3 cdot 2 equiv 6 equiv 1  (text{mod}  5)

Conclusion

Fields are the cornerstone of modern algebraic structures, used extensively in fields as diverse as number theory, algebraic geometry, and cryptography. By understanding fields, you gain insight into the symmetries and arrangements inherent in various mathematical systems, enabling you to exploit their deeper properties in both theoretical and practical applications.


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