Course Name: Galois Theory

Course abstract

Galois Theory is showpiece of a mathematical unification which brings together several different branches of the subject and creating a powerful machine for the study problems of considerable historical and mathematical importance. This course is an attempt to present the theory in such a light, and in a manner suitable for undergraduate and graduate students as well as researchers. This course will begin at the beginning. The quadratic formula for solving polynomials of degree 2 has been known for centuries and is still an important part of mathematics education. The corresponding formulas for solving polynomials of degrees 3 and 4 are less familiar. These expressions are more complicated than their quadratic counterpart, but the fact that they exist comes as no surprise. It is therefore altogether unexpected that no such formulas are available for solving polynomials of degree ≥ 5. A complete answer to this intriguing problem is provided by Galois theory. In fact Galois theory was created precisely to address this and related questions about polynomials. The participant is expected to have a basic knowledge of linear algebra, but other that the course is largely self-contained. Most of what is needed from fields and elementary theory polynomials is presented in the early lectures and much of the necessary group theory is also presented on the way. Classical notions, statements and their proofs are provided in modern set-up. Numerous examples are given to illustrate abstract notions. These examples are sort of an airport beacon, shining a clear light at our destination as we navigate a course through the mathematical skies to get there. Formally we cover the following topics : • Galois extensions and Fundamental theorem of Galois Theory. • Finite Fields, Cyclic Groups, Roots of Unity, Cyclotomic Fields. • Splitting fields, Algebraic closure • Normal and Separable extensions • Solvability of equations. • Inverse Galois Problem


Course Instructor

Media Object

Prof.Dilip Patil

Dilip P. Patil received B. Sc. and M. Sc. in Mathematics from the University of Pune in 1976 and 1978, respectively. From 1979 till 1992 he studied Mathematics at School of Mathematics, Tata Institute of Fundamental Research, Bombay and received Ph. D. through University of Bombay in 1989. Currently he is a Professor of Mathematics at the Departments of Mathematics, Indian Institute of Science, Bangalore. At present he is a Visiting Professor at the Department of Mathematics, IIT Bombay. He has been a Visiting Professor at Ruhr-Universität Bochum, Universität Leipzig, Germany and several universities in Europe and Canada. His research interests are mainly in Commutative Algebra and Algebraic Geometry
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Teaching Assistant(s)

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 Course Duration : Jul-Oct 2021

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 Syllabus

 Enrollment : 20-May-2021 to 02-Aug-2021

 Exam registration : 17-Jun-2021 to 17-Sep-2021

 Exam Date : 24-Oct-2021

Enrolled

611

Registered

11

Certificate Eligible

3

Certified Category Count

Gold

0

Silver

0

Elite

1

Successfully completed

2

Participation

5

Success

Elite

Silver

Gold





Legend

AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75 AND FINAL SCORE >=40
BASED ON THE FINAL SCORE, Certificate criteria will be as below:
>=90 - Elite + Gold
75-89 -Elite + Silver
>=60 - Elite
40-59 - Successfully Completed

Final Score Calculation Logic

  • Assignment Score = Average of best 8 out of 12 assignments.
  • Final Score(Score on Certificate)= 75% of Exam Score + 25% of Assignment Score
Galois Theory - Toppers list

AMEYA MOHAN BORWANKAR 72%

Institute Of Chemical Technology

Enrollment Statistics

Total Enrollment: 611

Registration Statistics

Total Registration : 11

Assignment Statistics




Assignment

Exam score

Final score

Score Distribution Graph - Legend

Assignment Score: Distribution of average scores garnered by students per assignment.
Exam Score : Distribution of the final exam score of students.
Final Score : Distribution of the combined score of assignments and final exam, based on the score logic.