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Course11110615186 lectures

Algebra - II

Prof. Sankaran Viswanath Prof. Amritanshu Prasad · IMSc

Start with lecture 1 At NPTEL

Lectures

Week 1

  1. Algebraic and Transcendental Numbers Video
  2. Extensions Generated by Elements Video
  3. Isomorphic Extensions Video
  4. Degree of an Extension Video
  5. Constructible Numbers Video
  6. The Field of Constructible Numbers Video
  7. Characterization of Constructible Numbers Video
  8. Solved Problems (Week 1) Video

Week 2

  1. Some Things can't be Constructed Video
  2. Symbolic Adjunction Video
  3. Repeated Roots Video
  4. Gauss Lemma Video
  5. Eisenstein's criterion Video
  6. Existence Theorem for Finite Fields Video
  7. Subfields of a Finite Field Video
  8. Multiplicative Group of a Finite Field Video
  9. Uniqueness Theorem for Finite Fields Video
  10. Solved Problems (week 2) Video

Week 3

  1. Algebraic Extensions and Algebraic Closures Video
  2. Existence of Algebraic Closures Video
  3. Uniqueness of Algebraic Closure Video
  4. Solved Problems, part 1 (Week 3) Video
  5. Existence of splitting fields, bound on degree Video
  6. Uniqueness of splitting fields Video
  7. Solved problems , part 2 (Week 3) Video

Week 4

  1. Normal Extensions Video
  2. Separable polynomials Video
  3. Perfect fields, separable extensions Video
  4. Definition and examples, fixed fields Video
  5. Characterization of Galois extensions Video
  6. Linear Independence of Characters Video
  7. Solved problems (Week 4) Video

Week 5

  1. Artin's Theorem - part 1 Video
  2. Artin's Theorem - part 2 Video
  3. Finite Galois Extensions Video
  4. The fundamental theorem of Galois Theory - 1 Video
  5. The fundamental theorem of Galois Theory - 2 Video
  6. Solved problems (Week 5) Video

Week 6

  1. Cyclotomic extensions Video
  2. Irreducibility of the cyclotomic polynomial Video
  3. Application: Constructibility of regular n-gons. Video
  4. Insolvability of the general quintic - part 1 Video
  5. Insolvability of the general quintic - part 2 Video
  6. Insolvability of the general quintic - part 3 Video

Week 7

  1. What is category theory (and why is it important)? Video
  2. Definition of a category Video
  3. Monomorphisms, epimorphisms, and isomorphisms Video
  4. Categories: First Problem Session Video
  5. Initial and Terminal Objects Video
  6. Products and Coproducts Video
  7. Categories: Second Problem Session Video

Week 8

  1. Functors Video
  2. The Category of Categories Video
  3. Natural Transformations Video
  4. Functor Categories Video
  5. Categories: Third Problem Session Video
  6. Adjunction Video
  7. Categories: Fourth Problem Session Video

Week 9

  1. Tensor products of Z-modules Video
  2. Free abelian groups and quotient groups Video
  3. Construction of the tensor product Video
  4. Problem session Video
  5. Tensor product of R-modules Video

Week 10

  1. Functoriality of the tensor product Video
  2. Bimodules Video
  3. Tensor products of bimodules Video
  4. Tensor products of modules over commutative rings Video
  5. Extension of scalars Video
  6. Problem session - tensor products of vector spaces Video

Week 11

  1. Some Properties of the tensor product Video
  2. F-algebras Video
  3. Composition Series Video
  4. Schreier's Theorem Video
  5. Ascending and Descending Chain Conditions Video
  6. Existence of Jordan-Holder Series Video
  7. The Jordan-Holder Theorem Video
  8. Examples related to the Jordan-Holder Theorem Video

Week 12

  1. The Jordan-Holder Theorem for Groups Video
  2. Indecomposable Modules Video
  3. Direct Sum Decompositions Video
  4. Decomposition as a sum of Indecomposables Video
  5. The Endomorphism Ring of an Indecomposable Module Video
  6. Krull-Schmidt Theorem Video
  7. Krull-Schmidt Examples Video

Live Session

  1. Live Session 05-03-2021 Video
  2. Live Session 03-04-2021 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IMSc
Department
Mathematics
Teacher
Prof. Sankaran Viswanath Prof. Amritanshu Prasad
Level
Undergraduate
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.
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