(i) Syllabus on Algebra and Number Theory
Algebra:
Group theory: Sylow theorems, p-groups, solvable groups, free
groups.
Rings and modules: tensor products, determinants, Jordan
canonical form, PID's, UFD's, polynomials rings.
Field theory: splitting fields, separable and inseparable extensions.
Galois theory: Fundamental theorems of Galois theory, finite fields,
cyclotomic fields.
Homological algebra: exact sequences, splittings, snake and five
lemmas, projective, injective, and flat modules, complexes,
(co)homology.
Commutative ring: localizations, Hilbert’s basis theorem, integral
extensions, radicals of ideals, Zariski topology and Hilbert’s
Nullstellensatz, Dedekind rings, DVRs.
Representations of Finite Groups: character theory, induced
representations, structure of the group ring.
Basics of Lie groups and Lie algebras: exponential map, nilpotent
and semi-simple Lie algebras and Lie groups.
References: Dummit and Foote: Abstract Algebra, 2nd edition; Serre:
Representations of Finite Groups; Fulton-Harris: Representation Theory:
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