Algebra

This is a algebra note made by a computer science student in case of his forgetting what he has learned. It might seem clumsy and not rigorous but should be plain enough for him to recall them once he has read the corresponding article.

Preliminaries

• Greatest common divisor
  • Euclidean algorithm
  • Least common multiple
  • Linear Diophantine equations
  • Chinese remainder theorem
• Prime
  • Fermat's little theorem
  • Euler's theorem
  • Primality test

Group Theory

• Groups
  • Groups
  • Subgroups
  • Quotient groups and normal subgroups
• Group homomorphisms
  • Group homomorphisms
  • First isomorphism theorem
  • Direct products
  • Second isomorphism theorem
  • Third isomorphism theorem
• Cyclic and finite abelian groups
  • Cyclic groups
  • Cyclic subgroups
  • Finite abelian groups
• Permutation groups
  • Permutation groups
  • Cycle notation
  • Alternating groups
• Application: Algebraic equation under permutation
• Group actions
  • Group actions
  • The class equation
  • Cauchy's theorem
  • The Sylow theorems
• Application: Polya's counting theory
  • Burnside's lemma
  • Polya's counting theory

Ring Theory

• Rings
  • Rings
  • Zero divisors and units
  • Characteristic of rings
  • Wedderburn's little theorem
• Ring homomorphisms and ideals
  • Ring homomorphisms
  • Principal ideals
  • Maximal ideals
  • Prime ideals
  • Chinese remainder theorem for rings
• Polynomial rings
  • Polynomial rings
  • (Ir)reducibility of polynomials
  • Existence theorem of primitive roots
  • Polynomials with integer coefficients
• Fractions
  • Axioms of fractions
  • Fields of fractions
  • Rings of fractions
• Vector spaces and linear transformations
  • Vector spaces
  • Linear transformations
  • Matrices of linear transformations
• Unique factorization domains
  • Unique factorization domains
  • Principal ideal domains
  • Euclidean domains
  • Polynomial rings over UFDs
• Application: Number theoretic functions and Dirichlet ring
  • Dirichlet ring
  • Möbius inversion formula
  • Generalized convolution formula

Fields and Galois Theory

• Field extensions and finite fields
  • Field extensions
  • Splitting fields
  • Finite fields
• Galois theory
  • Field automorphisms
  • Automorphism groups of finite extensions
  • Fixed fields
  • Normal and separable extensions
  • Fundamental theorem of Galois theory
• Solvability by radicals
  • Cyclotomic extensions
  • Simple radical extensions
  • Radical extensions and solvable groups
  • Calculation of Galois groups