# Introduction to Galois Theory

By Prof. Krishna Hanumanthu   |   Chennai Mathematical Institute
Learners enrolled: 77
In this introductory course on Galois theory, we will first review basic concepts from rings and fields, such as polynomial rings, field extensions and splitting fields. We will then learn about normal and separable extensions before defining Galois extensions. We will see a lot of examples and constructions of Galois groups and Galois extensions. We will then prove the fundamental theorem of Galois theory which gives a correspondence between subgroups of the Galois group and intermediate fields of a Galois extension. We will then cover some important applications of Galois theory, such as insolvability of quintics, Kummer extensions, cyclotomic extensions.
This course will focus a lot on solving exercises and giving plenty of examples. We will give several exercises to be done by students and will have weekly problem solving sessions where we will solve problems in detail.

INTENDED AUDIENCE:
Final year B.Sc students or M.Sc students in mathematics.

PREREQUISITES: Courses in linear algebra, group theory, rings and fields are prerequisites.
Summary
 Course Status : Upcoming Course Type : Elective Duration : 8 weeks Start Date : 23 Jan 2023 End Date : 17 Mar 2023 Exam Date : 26 Mar 2023 IST Enrollment Ends : 30 Jan 2023 Category : Mathematics Algebra Credit Points : 2 Level : Undergraduate/Postgraduate

### Course layout

Week 1:Review of rings and fields I: polynomial rings, irreducibility criteria, algebraic elements, field extensions
Week 2:Review of rings and fields II: finite fields, splitting fields
Week 3: Normal extensions, separable extensions
Week 4:Fixed fields, Galois groups
Week 5:Galois extensions, properties and examples
Week 6:Fundamental theorem of Galois theory
Week 7:Solvability by radicals, insolvability of quintics
Week 8:Kummer extensions, abelian extensions, cyclotomic extensions

### Books and references

Michael Artin: Algebra
Emile Artin: Galois Theory

### Prof. Krishna Hanumanthu

Chennai Mathematical Institute
Krishna Hanumanthu is an associate professor of mathematics at Chennai Mathematical Institute (CMI). He studied BSc and MSc in CMI during 1998-2003 and did his PhD in mathematics at University of Missouri during 2003-2008. He joined CMI as a faculty member in 2011 after working for 3 years at University of Kansas. His main areas of research are algebraic geometry and commutative algebra. He has been teaching for almost 15 years and taught introductory courses on abstract algebra (including group theory) many times.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams:
26 March 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 6 assignments out of the total 8 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team