Media Storage Type : 32 GB USB Stick
NPTEL Subject Matter Expert : Prof. Amritanshu Prasad, Prof. S. Viswanath
NPTEL Co-ordinating Institute : Institute of Mathematical Sciences
NPTEL Lecture Count : 84
NPTEL Course Size : 4.3 GB
NPTEL PDF Text Transcription : Available and Included
NPTEL Subtitle Transcription : Available and Included (SRT)
Lecture Titles:
Lecture 1 - Algebraic and Transcendental Numbers
Lecture 2 - Extensions Generated by Elements
Lecture 3 - Isomorphic Extensions
Lecture 4 - Degree of an Extension
Lecture 5 - Constructible Numbers
Lecture 6 - The Field of Constructible Numbers
Lecture 7 - Characterization of Constructible Numbers
Lecture 8 - Solved Problems (Week 1)
Lecture 9 - Some Things can't be Constructed
Lecture 10 - Symbolic Adjunction
Lecture 11 - Repeated Roots
Lecture 12 - Gauss Lemma
Lecture 13 - Eisenstein’s criterion
Lecture 14 - Existence Theorem for Finite Fields
Lecture 15 - Subfields of a Finite Field
Lecture 16 - Multiplicative Group of a Finite Field
Lecture 17 - Uniqueness Theorem for Finite Fields
Lecture 18 - Solved Problems (Week 2)
Lecture 19 - Algebraic Extensions and Algebraic Closures
Lecture 20 - Existence of Algebraic Closures
Lecture 21 - Uniqueness of Algebraic Closure
Lecture 22 - Solved Problems - Part 1 (Week 3)
Lecture 23 - Existence of splitting fields, bound on degree
Lecture 24 - Uniqueness of splitting fields
Lecture 25 - Solved problems - Part 2 (Week 3)
Lecture 26 - Normal Extensions
Lecture 27 - Separable polynomials
Lecture 28 - Perfect fields, separable extensions
Lecture 29 - Definition and examples, fixed fields
Lecture 30 - Characterization of Galois extensions
Lecture 31 - Linear Independence of Characters
Lecture 32 - Solved problems (Week 4)
Lecture 33 - Artin’s Theorem - Part 1
Lecture 34 - Artin’s Theorem - Part 2
Lecture 35 - Finite Galois Extensions
Lecture 36 - The fundamental theorem of Galois Theory - 1
Lecture 37 - The fundamental theorem of Galois Theory - 2
Lecture 38 - Solved problems (Week 5)
Lecture 39 - Cyclotomic extensions
Lecture 40 - Irreducibility of the cyclotomic polynomial
Lecture 41 - Application: Constructibility of regular n-gons.
Lecture 42 - Insolvability of the general quintic - Part 1
Lecture 43 - Insolvability of the general quintic - Part 2
Lecture 44 - Insolvability of the general quintic - Part 3
Lecture 45 - What is category theory (and why is it important)?
Lecture 46 - Definition of a category
Lecture 47 - Monomorphisms, epimorphisms, and isomorphisms
Lecture 48 - Categories: First Problem Session
Lecture 49 - Initial and Terminal Objects
Lecture 50 - Products and Coproducts
Lecture 51 - Categories: Second Problem Session
Lecture 52 - Functors
Lecture 53 - The Category of Categories
Lecture 54 - Natural Transformations
Lecture 55 - Functor Categories
Lecture 56 - Categories: Third Problem Session
Lecture 57 - Adjunction
Lecture 58 - Categories: Fourth Problem Session
Lecture 59 - Tensor products of Z-modules
Lecture 60 - Free abelian groups and quotient groups
Lecture 61 - Construction of the tensor product
Lecture 62 - Problem session
Lecture 63 - Tensor product of R-modules
Lecture 64 - Functoriality of the tensor product
Lecture 65 - Bimodules
Lecture 66 - Tensor products of bimodules
Lecture 67 - Tensor products of modules over commutative rings
Lecture 68 - Extension of scalars
Lecture 69 - Problem session - tensor products of vector spaces
Lecture 70 - Some Properties of the tensor product
Lecture 71 - F-algebras
Lecture 72 - Composition Series
Lecture 73 - Schreier’s Theorem
Lecture 74 - Ascending and Descending Chain Conditions
Lecture 75 - Existence of Jordan-Holder Series
Lecture 76 - The Jordan-Holder Theorem
Lecture 77 - Examples related to the Jordan-Holder Theorem
Lecture 78 - The Jordan-Holder Theorem for Groups
Lecture 79 - Indecomposable Modules
Lecture 80 - Direct Sum Decompositions
Lecture 81 - Decomposition as a sum of Indecomposables
Lecture 82 - The Endomorphism Ring of an Indecomposable Module
Lecture 83 - Krull-Schmidt Theorem
Lecture 84 - Krull-Schmidt Examples