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Core Module in Algebra

Associate Professor Peter Bruyns
1st semester 2020
20 credits / 30 lectures

  1. Abelian Groups: direct sums/products, torsion/torsion-free/mixed groups, independent subsets (Steinitz’ Theorem) and rank, groups of rank 1, finitely generated groups and their classification.
  2. Rings: Homomorphisms of rings, the quotient of a ring by an ideal, rings of polynomials, Integral Domains, Fields of Fractions, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains.
  3. Fields: characteristic of a field, extensions of fields, construction of extensions which contain certain desired elements, algebraic and transcendental elements over a field, finite fields.
  4. Vector Spaces: Dual Spaces, the double dual, polynomials associated with matrices/linear maps, Canonical Forms: triangular form, cyclic spaces and the Jordan canonical form for nilpotent matrices/maps, the Jordan Canonical Form for more general maps, the Exponent of a matrix, Bilinear Forms (symmetric and skew-symmetric forms).

Prerequisites:

The undergraduate module 3AL (Modern Abstract Algebra), or equivalent, is required. The 3rd year module 3TA (Topics in Algebra) is recommended.