Lecture 86 of 86Live Session
Live Session 03-04-2021
Prof. Sankaran Viswanath Prof. Amritanshu Prasad · IMSc
Credits and licence
- From
- Algebra - II, 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
- Algebraic and Transcendental Numbers
- Extensions Generated by Elements
- Isomorphic Extensions
- Degree of an Extension
- Constructible Numbers
- The Field of Constructible Numbers
- Characterization of Constructible Numbers
- Solved Problems (Week 1)
- Some Things can't be Constructed
- Symbolic Adjunction
- Repeated Roots
- Gauss Lemma
- Eisenstein's criterion
- Existence Theorem for Finite Fields
- Subfields of a Finite Field
- Multiplicative Group of a Finite Field
- Uniqueness Theorem for Finite Fields
- Solved Problems (week 2)
- Algebraic Extensions and Algebraic Closures
- Existence of Algebraic Closures
- Uniqueness of Algebraic Closure
- Solved Problems, part 1 (Week 3)
- Existence of splitting fields, bound on degree
- Uniqueness of splitting fields
- Solved problems , part 2 (Week 3)
- Normal Extensions
- Separable polynomials
- Perfect fields, separable extensions
- Definition and examples, fixed fields
- Characterization of Galois extensions
- Linear Independence of Characters
- Solved problems (Week 4)
- Artin's Theorem - part 1
- Artin's Theorem - part 2
- Finite Galois Extensions
- The fundamental theorem of Galois Theory - 1
- The fundamental theorem of Galois Theory - 2
- Solved problems (Week 5)
- Cyclotomic extensions
- Irreducibility of the cyclotomic polynomial
- Application: Constructibility of regular n-gons.
- Insolvability of the general quintic - part 1
- Insolvability of the general quintic - part 2
- Insolvability of the general quintic - part 3
- What is category theory (and why is it important)?
- Definition of a category
- Monomorphisms, epimorphisms, and isomorphisms
- Categories: First Problem Session
- Initial and Terminal Objects
- Products and Coproducts
- Categories: Second Problem Session
- Functors
- The Category of Categories
- Natural Transformations
- Functor Categories
- Categories: Third Problem Session
- Adjunction
- Categories: Fourth Problem Session
- Tensor products of Z-modules
- Free abelian groups and quotient groups
- Construction of the tensor product
- Problem session
- Tensor product of R-modules
- Functoriality of the tensor product
- Bimodules
- Tensor products of bimodules
- Tensor products of modules over commutative rings
- Extension of scalars
- Problem session - tensor products of vector spaces
- Some Properties of the tensor product
- F-algebras
- Composition Series
- Schreier's Theorem
- Ascending and Descending Chain Conditions
- Existence of Jordan-Holder Series
- The Jordan-Holder Theorem
- Examples related to the Jordan-Holder Theorem
- The Jordan-Holder Theorem for Groups
- Indecomposable Modules
- Direct Sum Decompositions
- Decomposition as a sum of Indecomposables
- The Endomorphism Ring of an Indecomposable Module
- Krull-Schmidt Theorem
- Krull-Schmidt Examples
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