Algebra II Course | Postgraduate Math | Galois Theory, Category Theory | IMSc
Course Details
| Exam Registration | 23 |
|---|---|
| Course Status | Ongoing |
| Course Type | Core |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 17 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Algebra - II: A Foundational Postgraduate Course in Advanced Algebra
Building upon the core principles established in Algebra I, Algebra - II is an essential 12-week postgraduate course designed to delve into the profound structures of modern algebra. Taught by distinguished faculty from The Institute of Mathematical Sciences (IMSc), Chennai, this course is meticulously crafted for M.Sc. and first-year Ph.D. students aiming to solidify their mathematical foundation.
Course Instructors
This course is led by two eminent researchers in representation theory:
- Prof. Amritanshu Prasad: A faculty member at IMSc Chennai, whose research expertise provides deep insight into the algebraic structures covered in this course.
- Prof. S. Viswanath: Also a faculty member at IMSc Chennai, bringing his specialized knowledge in representation theory to the curriculum.
Who Should Take This Course?
This is an advanced, postgraduate-level course. A strong grasp of topics from a foundational Algebra I course (covering groups, rings, and modules) is highly recommended as a prerequisite. It is perfectly suited for:
- M.Sc. students in Mathematics seeking advanced specialization.
- First-year Ph.D. students beginning research in pure mathematics, particularly algebra and related fields.
- Researchers and professionals looking to refresh or deepen their understanding of core algebraic concepts.
Detailed 12-Week Course Layout
The course progresses from field theory and classical problems to modern abstract structures, providing a comprehensive journey through advanced algebra.
| Week | Topic | Key Concepts |
|---|---|---|
| Week 1 | Fields, Equations, Extensions | Field extensions, algebraic vs. transcendental, minimal polynomials. |
| Week 2 | Ruler and Compass Constructions | Classical problems (squaring the circle, doubling the cube), connection to field theory. |
| Week 3 | Finite Fields | Structure, existence, and uniqueness of finite fields (Galois fields). |
| Week 4 | Galois Theory - 1 | Galois groups, fundamental theorem of Galois theory (beginning). |
| Week 5 | Galois Theory - 2 | Applications: solvability by radicals, insolvability of the quintic. |
| Week 6 | Categories, Functors, Natural Transformations - 1 | Introduction to the language of category theory. |
| Week 7 | Categories, Functors, Natural Transformations - 2 | Universal properties, limits, and colimits. |
| Week 8 | Tensor Products of Modules | Construction, universal property, and basic properties. |
| Week 9 | Jordan-Holder Theorem | Composition series, simple modules, uniqueness of factors. |
| Week 10 | Krull-Schmidt Theorem | Indecomposable modules, uniqueness of direct sum decompositions. |
| Week 11 | Semisimple Rings, Artin-Wedderburn Theorem | Structure theory of semisimple rings and algebras. |
| Week 12 | Multilinear Algebra | Exterior and symmetric algebras, determinants from a advanced perspective. |
Core Learning Outcomes
By completing Algebra - II, students will achieve a robust understanding of:
- The powerful correspondence between field extensions and groups in Galois Theory, solving centuries-old problems.
- The unifying language of Category Theory, essential for modern mathematics and theoretical computer science.
- The construction and properties of Tensor Products, a cornerstone of advanced algebra, geometry, and representation theory.
- Fundamental structure theorems for modules and rings: the Jordan-Holder and Krull-Schmidt theorems.
- The complete classification of semisimple rings via the celebrated Artin-Wedderburn Theorem.
Recommended Textbooks
To support your learning journey, the following texts are highly recommended:
- Algebra by Michael Artin
- Abstract Algebra by David S. Dummit and Richard M. Foote
- Basic Category Theory for Computer Scientists by Benjamin C. Pierce (for the category theory weeks)
This course, designed and delivered by active researchers, bridges the gap between advanced undergraduate studies and the frontier of mathematical research. It equips students with the sophisticated algebraic tools necessary for further study in representation theory, algebraic geometry, number theory, and beyond.
Enroll Now →