Abstract Algebra: Fields

• Algebra deals with symbols and rules for manipulating them.
• Symbols represent numbers or quantities.
• Algebra is a generalization of arithmetic.
• Elementary algebra focuses on solving equations with variables.
• Abstract algebra extends concepts from elementary algebra and arithmetic.

Fields

• A field is a fundamental algebraic structure.
• It is a set where addition, subtraction, multiplication, and division are defined.
• Operations behave like those on rational and real numbers.
• Common fields include real numbers, complex numbers, and rational numbers.

Definition of a Field

• A field is a set F with addition and multiplication as binary operations.
• A binary operation from a set to itself associates a unique element of the set.
• The addition of a and b is a + b.
• The multiplication of a and b is a * b or ab.
• Operations must satisfy specific axioms.

Field Axioms

• Associativity of addition: (a + b) + c = a + (b + c) for all a, b, c in F.
• Commutativity of addition: a + b = b + a for all a, b in F.
• Additive identity: There exists 0 in F such that a + 0 = a for all a in F.
• Additive inverse: For every a in F, there exists -a in F such that a + (-a) = 0.
• Associativity of multiplication: (a * b) * c = a * (b * c) for all a, b, c in F.
• Commutativity of multiplication: a * b = b * a for all a, b in F.
• Multiplicative identity: There exists 1 in F such that 1 ≠ 0 and a * 1 = a for all a in F.
• Multiplicative inverse: For every a ≠ 0 in F, there exists a⁻¹ in F such that a * a⁻¹ = 1.
• Distributivity of multiplication over addition: a * (b + c) = (a * b) + (a * c) for all a, b, c in F.

Properties of Fields

• Additive and multiplicative identities are unique.
• Every element has a unique additive inverse.
• Every nonzero element has a unique multiplicative inverse.
• Zero Product Property: if a * b = 0, then either a = 0 or b = 0 (or both).

Examples of Fields

• Rational Numbers (Q): Numbers expressed as p/q, where p and q are integers and q ≠ 0.
• Real Numbers (R): Numbers represented on a number line.
• Complex Numbers (C): Numbers of the form a + bi, where a and b are real numbers, and i² = -1.
• Finite Fields: Fields with a finite number of elements, also known as Galois fields.

Non-Examples of Fields

• Integers (Z): Whole numbers and their negatives are not a field as not every nonzero integer has a multiplicative inverse.
• Natural Numbers (N): Positive whole numbers are not a field because they lack additive inverses and identities as well as multiplicative inverses.

Applications of Fields

• Cryptography: Finite fields are used in cryptographic algorithms.
• Coding Theory: Fields are used to construct error-correcting codes.
• Physics: Real and complex fields are used extensively in physics.
• Computer Science: Fields play a role in various algorithms and data structures.
• Abstract Algebra: Fields are fundamental to the study of algebraic structures.

Finite Fields

• A finite field is a field with a finite number of elements.
• The number of elements is always a prime power p^n (p is prime, n is a positive integer).
• Finite fields are denoted as GF(p^n) (GF stands for Galois Field).
• When n = 1, the field is denoted as Z/pZ or F_p (integers modulo p).
• Arithmetic in finite fields is performed modulo p.
• GF(2) is the simplest finite field with two elements: 0 and 1.

Importance of Fields in Algebra

• Fields provide a foundation for vector spaces, rings, and modules.
• Theorems in algebra rely on field axioms.
• Fields allow the application of algebraic techniques in mathematics.

Subfields

• A subfield of F is a subset of F that is itself a field.
• To check if K is a subfield of F:
  • K must contain 0 and 1.
  • For any a, b in K, a - b is in K.
  • For any a, b in K with b ≠ 0, a * b⁻¹ is in K.

Field Extensions

• A field extension is a field F containing a smaller field E as a subfield.
• Field extensions are used to construct new fields.
• The degree of a field extension F/E, denoted [F:E], is the dimension of F as a vector space over E.