Abstract Algebra

1. Fields
   1. Definition and Examples
      1. Definition of a Field
         1. Two operations: addition and multiplication
         2. Inverses for both operations
         3. Distributive law linking the two operations
      2. Examples of Fields
         1. Rational Numbers (Q)
         2. Real Numbers (R)
         3. Complex Numbers (C)
         4. Finite fields, often denoted as F_p
   2. Field Axioms
      1. Commutative Law
         1. Addition: a + b = b + a
         2. Multiplication: a * b = b * a
      2. Associative Law
         1. Addition: (a + b) + c = a + (b + c)
         2. Multiplication: (a * b) * c = a * (b * c)
      3. Identity Element
         1. Addition: exists an element 0 such that a + 0 = a
         2. Multiplication: exists an element 1 (≠ 0) such that a * 1 = a
      4. Inverses
         1. Additive Inverse: for every a, exists an element -a such that a + (-a) = 0
         2. Multiplicative Inverse: for every a ≠ 0, exists an element a^(-1) such that a * a^(-1) = 1
      5. Distributive Property
         1. a * (b + c) = a * b + a * c
   3. Subfields and Extensions
      1. Subfields
         1. Definition and examples
         2. Criteria for subfields
         3. Subfield lattice structure
      2. Field Extensions
         1. Definition and examples
         2. Types of extensions
      3. Simple Extensions
         1. Definition and construction
         2. Examples such as Q(√2)
         3. Minimal polynomials
      4. Algebraic Extensions
         1. Algebraic element definition
         2. Degree of an extension
         3. Finite extensions and the Tower Law
      5. Transcendental Extensions
         1. Transcendental elements definition
         2. Examples: Extension by π or e
         3. Construction and properties
   4. Finite Fields
      1. Construction and Properties
         1. Existence of finite fields of size p^n (p prime)
         2. Construction via polynomial rings
      2. Structure Theorems
         1. Uniqueness of finite fields up to isomorphism
         2. Multiplicative group of nonzero elements being cyclic
      3. Applications in Cryptography
         1. Use in designing cryptographic algorithms
         2. Error-detecting and error-correcting codes
         3. Diffie-Hellman and other public key cryptosystems
   5. Field Homomorphisms
      1. Definition and Examples
         1. Structure-preserving maps between fields
         2. Kernel of a field homomorphism
      2. Embeddings
         1. Injective homomorphisms
         2. Examples such as embedding Q into R
      3. Automorphisms
         1. Field isomorphisms from a field to itself
         2. Galois groups as sets of automorphisms
      4. Fixed Fields
         1. Fixed elements under a set of automorphisms
         2. Field of invariants in Galois theory