Algebra I

1. Groups
   1. Examples, category of groups, Action of a group on a set.
   2. Subgroups, isomorphism theorems.
   3. Group actions: Permutation representations, action on itself by left multiplication, action on itself by conjugation.
2. Automorphisms of groups and statement of Sylow theorem
   1. Automorphisms: Inner automorphisms, automorphism groups of some finite groups: dihedral, quaternions, cyclic.
   2. Statement of Sylow’s theorem, Direct and Semidirect products.
   3. Simple groups, composition series, Jordan-Hölder Series, A_n is simple.
3. Category Theory
   1. Objects, morphisms, functors.
4. Free groups
   1. Free groups: words, construction, and uniqueness.
   2. Universal property, adjointness with forgetful functor.
   3. Finitely generated and finitely presented groups.
5. Rings
   1. Definitions (review): integral domains, euclidean domains, pid, ufd, fields.
   2. Examples: Polynomials rings, Matrix rings, group rings.
   3. Ideals and Quotient rings, prime and maximal ideals.
   4. Chinese Reminder Theorem.
   5. Nilradical and Jacobson radical.
6. Modules
   1. Definition, Z-modules, F[x]-modules.
   2. Direct sums and free modules - construction and universal property.
7. Bilinear Forms
   1. Symmetric forms. Orthogonal bases, ordered fields, Gram Schmidt, Sylvester’s theorem.
   2. Eigen vectors of linear maps, Spectral theorem (Hermitian, Unitary, Symmetric case).
   3. Structure theorem for alternating forms.
8. Tensors
   1. Tensor products of modules. Examples.
   2. Universal property, Adjointness with Hom.
   3. Tensor product of homomorphisms, associativity, symmetry, tensor product of algebras.
9. Symmetric and Exterior algebras
   1. Linear functions on tensor products of vector spaces, determinants.
   2. Symmetric algebras, universal properties, alternating algebras, universal properties, symmetric and alternating tensors.
10. Modules over a PID and Canonical forms
    1. Structure of finitely generated modules over a PID.
    2. Canonical forms.
    3. Rational Canonical Form.
    4. Jordan Canoncial Form.

Algebra II

1. Field theory
   1. Characteristic of a field, extensions, degree of an extension, primitive elements for an extension.
   2. Algebraic extensions, finitely generated field extensions, compositum of two fields.
   3. Splitting fields and algebraic closure.
2. Separability
   1. Separable and Inseparable extensions.
   2. Fields of characteristic p > 0. Finite fields. Perfect fields.
   3. Separable and inseparable degrees.
   4. Primitive Element theorem.
3. Galois Theory
   1. Galois extensions and Galois groups.
   2. Linear independence of characters.
   3. Fundamental theorem of Galois Theory.
   4. Example: Cyclotomic extensions.
   5. Frobenius automorphism and Galois groups of finite fields.
   6. Normal basis theorem.
   7. Infinite Galois extensions.
   8. Krull topology on the Galois group and version of Fundamental theorem for infinite Galois extensions.
4. Modules and algebras
   1. Exact sequences of modules, tensor products, flatness and absolute flatness.
   2. Restriction and extension of scalars, tensor product of algebras
   3. Projective modules and Injective modules.
   4. Ext and Tor functors: Definitions and basic properties.
   5. Chain complexes of ℤ[G] modules and Group cohomology
   6. Hilbert's theorem 90
   7. Correspondence between H²(G,A) and extensions
5. Commutative Algebra
   1. Localization of rings and modules
      1. Localization.
      2. Universal property.
      3. Exactness of localization functor.
   2. Integral dependence
      1. Integral dependence.
      2. Going up lemma.
   3. Chain conditions
      1. Noetherian modules and rings.
      2. Hilbert Basis theorem.
      3. Artinian modules and rings.
   4. Spec of a ring and basics of Zariski topology.

References

• 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