Algebra I and II

Algebra I, II – 60210, 60220

  1. Groups
    Groups. Cyclic groups, permutation groups symmetric and alternating groups, matrix groups. Subgroups, quotient groups, direct products, homomorphism theorems. Automorphisms, conjugacy. Cosets, Lagrange's theorem. Group actions, G-sets, Sylow theorems, free groups and presentations.
  2. Rings
    Rings. Polynomial rings, matrix rings. Ideals, quotient rings, homomorphism theorems. Prime and maximal ideals. Principal ideal domains, Euclidean domains, unique factorization domains. Localization of rings, field of fractions.
  3. Modules
    Finite dimensional vector spaces. Basis, dimension, linear transformations, matrices. Modules. Submodules, quotient modules, homomorphism theorems, structure of finitely generated modules over a principal ideal domain. Language of categories and functors. Direct sums and products, free, projective and injective modules. Duality, multilinear forms, determinants.
  4. Canonical forms of matrices of linear transformations
    Jordan, rational and primary rational canonical forms. Invariant factors and elementary divisors.
  5. Fields
    Fields, algebraic and transcendental field extensions, degree, transcendence degree, algebraic closure. Fundamental theorem of Galois theory, separability, normality. Finite fields. *Cyclotomic extensions, *cyclic extensions, *solvable and nilpotent groups, *Impossibility proofs (trisecting angles, etc), *solvability of polynomial equations by radicals
  6. Tensor products
    Tensor products, *algebras, *(the tensor, symmetric and exterior algebras)
  7. Chain conditions
    Artinian and Noetherian rings and modules, Hilbert basis theorem. *Simple and semisimple module
  8. Commutative algebra
    Localization of modules. Integral ring extensions. *Localization and spec of a ring. *Nakayama's lemma, *Noether normalization lemma. *Nullstellensatz. *Affine algebraic varieties and commutative rings.


  • T. Hungerford, Algebra, Graduate Texts in Mathematics 73.
  • N. Jacobson, Basic algebra I, II. S. Lang, Algebra.