Course11110613783 lectures
Algebra - I
Prof. S. Viswanath and Prof. Amritanshu Prasad · IMSc
About
Foundational PG level course in algebra, suitable for M.Sc and first year PhD students in Mathematics.
Intended audience: Any Interested Learners
Prerequisites: BSc-level linear algebra
Lectures
Week 1
Week 2
- Group Actions
- Cosets
- Group Homomorphisms
- Normal subgroups
- Quotient Groups
- Product and Chinese Remainder Theorem
- Dihedral Groups
- Semidirect products
- Problem solving
Week 3
- The Orbit Counting Theorem
- Fixed points of group actions
- Second application: Fixed points of group actions
- Sylow Theorems - a preliminary proposition
- Sylow Theorem I
- Problem solving I
- Problem solving II
Week 4
- Sylow Theorem II
- Sylow Theorem III
- Problem solving I
- Problem solving II
- Free Groups I
- Free Groups IIa
Week 5
- Free Groups IIb
- Free Groups III
- Free Groups IV
- Problem Solving/Examples
- Generators and relations for symmetric groups – I
- Generators and relations for symmetric groups – II
Week 6
- Definition of a Ring
- Euclidean Domains
- Gaussian Integers
- The Fundamental Theorem of Arithmetic
- Divisibility and Ideals
- Factorization and the Noetherian Condition
- Examples of Ideals in Commutative Rings
- Problem Solving/Examples
Week 7
- The Ring of Formal Power Series
- Fraction Fields
- Path Algebra of a Quiver
- Ideals In Non-Commutative Rings
- Product of Rings
- Ring Homomorphisms
- Quotient Rings
- Problem solving
Week 8
- Tensor and Exterior Algebras
- Modules : definition
- Modules over polynomial rings $K[x]$
- Modules: alternative definition
- Modules: more examples
- Submodules
- General constructions of submodules
- Problem solving
Week 9
- Quotient modules
- Homomorphisms
- More examples of homomorphisms
- First isomorphism theorem
- Direct sums of modules
- Complementary submodules
- Change of ring
- Problem solving
Week 10
Week 11
- Finitely generated modules and the Noetherian condition
- Counterexamples to the Noetherian condition
- Generators and relations for Finitely Generated Modules
- General Linear Group over a Commutative Ring
- Equivalence of Matrices
- Smith Canonical Form for a Euclidean domain
- solved_problems1
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IMSc
- Department
- Mathematics
- Taught by
- Prof. S. Viswanath, Prof. Amritanshu Prasad
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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