Doctorate Algebra Syllabus

Groups

  • Series of groups (Jordan-Holder Theorem), solvable and nilpotent groups
  • Representation theory (Maschke's Theorem)

Rings

  • Modules
  • Division rings (Wedderburn's Theorem)
  • Noetherian rings (Hilbert Basis Theorem, power series rings), localization
  • Semi-simple rings (Density Theorem)
  • Gröbner Bases
  • Affine algebraic varieties

Fields

  • Separable, inseparable and purely inseparable extensions, normal extensions
  • Transcendental extensions (transcendence bases)
  • Galois theory

References

Dummit and Foote: Abstract Algebra
Hungerford: Algebra
Lang: Algebra
Rotman: Advanced Modern Algebra

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