Introduction to Galois Theory Course | CMI | Prof. Krishna Hanumanthu
Course Details
| Exam Registration | 19 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 8 weeks |
| Categories | Mathematics, Algebra |
| Credit Points | 2 |
| Level | Undergraduate/Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 13 Mar 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 16 Feb 2026 |
| Exam Date | 28 Mar 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Unlocking the Secrets of Equations: Your Guide to Galois Theory
Have you ever wondered why there's no general formula to solve polynomial equations of degree 5 or higher, unlike the quadratic formula we all learn in school? The answer lies in a beautiful and profound area of mathematics known as Galois Theory. This elegant framework, developed by the brilliant Évariste Galois in the 19th century, connects the world of field extensions with the world of groups, providing deep insights into the solvability of polynomial equations.
We are thrilled to present a detailed introductory course on this fascinating subject, taught by an expert in the field. This 8-week journey is designed to build your understanding from the ground up, culminating in some of the most celebrated results in algebra.
Meet Your Instructor: Prof. Krishna Hanumanthu
This course is led by Prof. Krishna Hanumanthu, an associate professor of mathematics at the prestigious Chennai Mathematical Institute (CMI). With a strong academic foundation from CMI (BSc, MSc 1998-2003) and a PhD from the University of Missouri (2003-2008), Prof. Hanumanthu brings over 15 years of teaching experience to the table.
His research expertise in algebraic geometry and commutative algebra provides a rich, deep perspective on the subject. Having taught introductory abstract algebra courses numerous times, he is exceptionally skilled at guiding students through complex concepts with clarity, focusing on examples and problem-solving—a cornerstone of this course.
Who Is This Course For?
This course is specifically designed for:
- Final year B.Sc. students in Mathematics.
- M.Sc. students in Mathematics or related fields.
Prerequisites: A solid understanding of Linear Algebra, Group Theory, and Rings & Fields is essential. Familiarity with a course like Introduction to Rings and Fields (such as the one available on NPTEL) is highly recommended to ensure you can fully engage with the material.
What Will You Learn? A Week-by-Week Breakdown
The course is structured over eight intensive weeks, each building upon the last to form a complete picture of classical Galois Theory.
| Week | Topics Covered |
|---|---|
| Week 1 & 2 | Review & Foundations: We begin by solidifying your understanding of polynomial rings, irreducibility criteria, field extensions, finite fields, and the crucial concept of splitting fields. |
| Week 3 | Key Properties: Dive into the characteristics that define well-behaved extensions: normal extensions and separable extensions. |
| Week 4 & 5 | The Heart of the Theory: Define and explore fixed fields, Galois groups, and Galois extensions through numerous examples and constructions. |
| Week 6 | The Core Theorem: Prove the Fundamental Theorem of Galois Theory, which establishes a perfect correspondence between subgroups of the Galois group and intermediate fields of the extension. |
| Week 7 | A Famous Application: Apply the theory to understand solvability by radicals and prove the monumental result: the insolvability of the general quintic equation. |
| Week 8 | Further Applications: Study important classes of extensions like Kummer extensions, abelian extensions, and cyclotomic extensions. |
Course Philosophy: Learning by Doing
This course emphasizes active learning. You won't just passively listen to lectures. The pedagogy includes:
- Plenty of Examples: Each abstract concept will be immediately illustrated with concrete examples.
- Dedicated Exercises: Weekly problem sets will challenge you to apply the concepts and deepen your understanding.
- Problem-Solving Sessions: Regular sessions where selected exercises and challenging problems will be solved in detail, fostering a collaborative learning environment.
Recommended Textbooks
To supplement your learning, the course will align with two classic and highly regarded texts:
- Michael Artin: Algebra – Excellent for a broad overview of algebra with a modern perspective.
- Emile Artin: Galois Theory – A concise, classic masterpiece focused directly on the subject.
Why Study Galois Theory?
Galois Theory is more than just a chapter in an algebra textbook. It is a cornerstone of modern mathematics, with applications reaching into number theory, algebraic geometry, and cryptography. By completing this course, you will:
- Gain a profound understanding of the structure of field extensions.
- Master the powerful dictionary between field theory and group theory.
- Appreciate the elegant proof behind one of mathematics' most famous impossibilities.
- Build a strong foundation for advanced studies in pure mathematics.
Join Prof. Krishna Hanumanthu on this 8-week intellectual adventure into the heart of abstract algebra. Whether your goal is to prepare for advanced research or simply to grasp one of mathematics' most beautiful theories, this course in Galois Theory is the perfect starting point.
Enroll Now →