Lecture 21 of 83Week 3
Problem solving II
Prof. S. Viswanath and Prof. Amritanshu Prasad · IMSc
Credits and licence
- From
- Algebra - I, IMSc (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- 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
- Tweens profiles and older. How courses are placed
All the lectures
- Permutations
- Group Axioms
- Order and Conjugacy
- Subgroups
- Problem solving
- Group Actions
- Cosets
- Group Homomorphisms
- Normal subgroups
- Quotient Groups
- Product and Chinese Remainder Theorem
- Dihedral Groups
- Semidirect products
- Problem solving
- 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
- Sylow Theorem II
- Sylow Theorem III
- Problem solving I
- Problem solving II
- Free Groups I
- Free Groups IIa
- 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
- 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
- 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
- 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
- Quotient modules
- Homomorphisms
- More examples of homomorphisms
- First isomorphism theorem
- Direct sums of modules
- Complementary submodules
- Change of ring
- Problem solving
- Free Modules (finitely generated)
- Determinants
- Primary Decomposition
- Problem solving
- 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
- Smith Canonical Form for PID
- Structure of finitely generated modules over a PID
- Structure of a finitely generated abelian group
- Similarity of Matrices
- Deciding Similarity
- Rational Canonical Form
- Jordan Canonical Form