Group Theory:
Review of basics: Groups, subgroups, cyclic groups, quotient groups, Lagrange’s theorem and some applications, isomorphism theorems, composition series,
Jordan-Holder theorem.
Group actions: Definition and examples, Cauchy’s theorem, class equation, Sylow’s theorems and applications.
Direct product: Definition and examples, structure theorem for finitely generated abelian groups, examples.
Solubility: Derived and lower central series, soluble groups, examples.
Free groups, group presentation, nilpotent groups (if time permits)
Rings and Modules:
Review of basics: Rings, subrings, ideals, examples, ring homomorphism, quotient rings, isomorphism theorems, field, integral domain, prime and maximal ideals,
characterization of prime and maximal ideals, direct product of rings, chinese remainder theorem.
Localization: Definition and examples, universal property of localization, local ring, localization at a prime ideal.
Integral domains: Euclidean domain, Principal ideal domain, Unique factorization domain, primes and irreducible elements, examples, Gauss’s lemma, Eisenstein’s criterion.
Polynomial rings: polynomial rings in one and several variables, universal property, unique factorization property.
Basics on modules: Modules, submodules, homomorphism of modules, isomorphism theorems, ring of endomorphisms of a module.
Structure theorems: Direct product of modules, direct sum of modules, universal property of direct product and direct sum of modules, short exact sequence, short five lemma,
structure theorem for finitely generated modules over a principal ideal domain (with proof), review the fundamental theorem of finitely generated abelian groups.
Canonical forms: Rational and Jordan canonical forms.
Fields and Galois Theory:
Review of basics: Fields, subfield, characteristic of a field, field homomorphism.
Field extensions: Finite and algebraic extensions, splitting fields, normal extensions, algebraic closure, separable extensions, inseparable extensions, cyclotomic fields,
finite fields.
Galois Theory: Primitive element theorem, fundamental theorem of Galois theory, applications, simple extensions.