- Groups
- Examples, category of groups, Action of a group on a set.
- Subgroups, isomorphism theorems.
- Group actions: Permutation representations, action on itself by left multiplication, action on itself by conjugation.

- Automorphisms of groups and statement of Sylow theorem
- Automorphisms: Inner automorphisms, automorphism groups of some finite groups: dihedral, quaternions, cyclic.
- Statement of Sylow’s theorem, Direct and Semidirect products.
- Simple groups, composition series, Jordan-Hölder Series, A
_{n}is simple.

- Category Theory
- Objects, morphisms, functors.

- Free groups
- Free groups: words, construction, and uniqueness.
- Universal property, adjointness with forgetful functor.
- Finitely generated and finitely presented groups.

- Rings
- Definitions (review): integral domains, euclidean domains, pid, ufd, fields.
- Examples: Polynomials rings, Matrix rings, group rings.
- Ideals and Quotient rings, prime and maximal ideals.
- Chinese Reminder Theorem.
- Nilradical and Jacobson radical.

- Modules
- Definition, Z-modules, F[x]-modules.
- Direct sums and free modules - construction and universal property.

- Bilinear Forms
- Symmetric forms. Orthogonal bases, ordered fields, Gram Schmidt, Sylvester’s theorem.
- Eigen vectors of linear maps, Spectral theorem (Hermitian, Unitary, Symmetric case).
- Structure theorem for alternating forms.

- Tensors
- Tensor products of modules. Examples.
- Universal property, Adjointness with Hom.
- Tensor product of homomorphisms, associativity, symmetry, tensor product of algebras.

- Symmetric and Exterior algebras
- Linear functions on tensor products of vector spaces, determinants.
- Symmetric algebras, universal properties, alternating algebras, universal properties, symmetric and alternating tensors.

- Modules over a PID and Canonical forms
- Structure of finitely generated modules over a PID.
- Canonical forms.
- Rational Canonical Form.
- Jordan Canoncial Form.

- Field theory
- Characteristic of a field, extensions, degree of an extension, primitive elements for an extension.
- Algebraic extensions, finitely generated field extensions, compositum of two fields.
- Splitting fields and algebraic closure.

- Separability
- Separable and Inseparable extensions.
- Fields of characteristic p > 0. Finite fields. Perfect fields.
- Separable and inseparable degrees.
- Primitive Element theorem.

- Galois Theory
- Galois extensions and Galois groups.
- Linear independence of characters.
- Fundamental theorem of Galois Theory.
- Example: Cyclotomic extensions.
- Frobenius automorphism and Galois groups of finite fields.
- Normal basis theorem.
- Infinite Galois extensions.
- Krull topology on the Galois group and version of Fundamental theorem for infinite Galois extensions.

- Modules and algebras
- Exact sequences of modules, tensor products, flatness and absolute flatness.
- Restriction and extension of scalars, tensor product of algebras
- Projective modules and Injective modules.
- Ext and Tor functors: Definitions and basic properties.
- Chain complexes of ℤ[G] modules and Group cohomology
- Hilbert's theorem 90
- Correspondence between H
^{2}(G,A) and extensions

- Commutative Algebra
- Localization of rings and modules
- Localization.
- Universal property.
- Exactness of localization functor.

- Integral dependence
- Integral dependence.
- Going up lemma.

- Chain conditions
- Noetherian modules and rings.
- Hilbert Basis theorem.
- Artinian modules and rings.

- Spec of a ring and basics of Zariski topology.

- Localization of rings and modules

- Dummit & Foote: Abstract Algebra.
- Hungerford: Algebra.
- Herstein: Abstract Algebra.
- Artin: Algebra.
- Lang: Algebra.
- Bourbaki: Algebra.
- Alperin & Bell: Groups and Representations.
- Atiyah & MacDonald: Introduction to Commutative Algebra.
- Bourbaki: Commutative Algebra.
- Weibel: Introduction to Homological Algebra.
- Jacobson: Basic Algebra I & II.

Uploaded on: June 8, 2020