Comps Syllabus for Algebra

Download algebra comps syllabus

Sets, Functions, and Integers

  • One-to-one, onto and bijective maps
  • Equivalence relations and equivalence classes
  • Division algorithm, gcd and lcm, primes and prime factorization

Groups

  • Uniqueness of identities and inverses
  • The order of an element

Subgroups

  • Lagrange's Theorem and its consequences
  • Cosets
  • Normal subgroups
  • Quotient groups

Group Homorphisms

  • Kernels and images
  • Isomorphisms
  • The Fundamental Theorem of Group Homomorphisms: G/Ker(φ)≅Im(φ)

Permutations

  • Sn and disjoint cycle decomposition
  • Transpositions and An

Rings

  • Commutative rings; Rings with unity; Fields
  • Polynomial rings

Ideals

  • Ideals
  • Quotient rings

Ring Homomorphisms

  • Kernels and images
  • Isomorphisms
  • The Fundamental Theorem of Ring Homomorphisms: R/Ker(φ)≅Im(φ)

Quotient Rings and Fields

  • Criteria for R to be a field
  • Maximal ideals
  • Criteria for R/M to be a field

Polynomial Rings k[x], for a Field k

  • The division algorithm
  • Every ideal in k[x] is principal
  • Irreducible polynomials and maximal ideals in k[x]