ISL Galois Theory (2014)

Venue: S.G.G.S. Institute of Engineering and Technology Nanded, Maharashtra
Dates: 1st to 13th Dec, 2014


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Sudhir R. Ghorpade Arunkumar R. Patil
Mailing Address Department of Mathematics,
Indian Institute of Technology Bombay,
Powai, Mumbai 400 076
S.G.G.S. Institute of Engineering and Technology,
Vishnupuri, Nanded 431 606


Speakers and Syllabus 

Resource Persons:
Five resource persons from reputed institutions of India delivered lectures in the school. These resource persons were as follows:
1. Prof. Sudhir R Ghorpade, IIT Mumbai.
2. Prof. Arunkumar R. Patil, SGGS IE and T, Nanded
3. Prof. Indranath Sengupta, IIT Gandhinagar.
4. Prof. Jugal K. Verma, IIT Mumbai.
5. Prof. Balwant Singh, CEBS, Mumbai.
These speakers together delivered 24 lectures of one-and-half hour duration spread over two weeks. In addition, four lectures on Unity in Mathematics were arranged. These lectures were given by following two eminent speakers:
1. Prof. V. Srinivas, School of Mathematics, TIFR, Mumbai had delivered a talk on Connection of Galois Theory with fundamental groups on December 11, 2014.
2. Prof. T. N. Venkataramana, School of Mathematics, TIFR, Mumbai had delivered a talk on connection of Galois Theory with monodromy groups on December 12, 2014.
Two hours tutorials were conducted on every day. Along with instructors following three persons assisted as a tutor to conduct tutorial sessions:
1. Mrinmoy Datta, Reserach Scholar, IIT Bombay, Mumbai
2. Joydip Saha, RKM Vivekananda University, Belur Math, Howrah, Kolkatta
3. Gaurab Tripathi, Research Scholar, Jadhavpur University

Week 1

December 1

December 2
December 3

December 4

December 5
December 6
9:30 - 11:00 SRG SRG SRG IS SRG IS
11:00 - 11:30 Tea/coffee
11:30 - 1:00 ARP ARP SRG IS IS IS
2:30 - 4:30 Tutorial
(SRG +    Tutors)
(SRG +    Tutors)
(ARP+    Tutors)
(IS +    Tutors)
(IS +    Tutors)
(IS +    Tutors)
4:30 - 5:00 Tea/ coffee/ Snacks


Week 2

December 8

December 9
December 10

December 11

December 12
December 13
9:00 - 10:30 JKV JKV BS JKV UiMT - VS BS
10:30 - 11:00 Tea/coffee
11:00 - 12:30 IS SRG JKV UiMT - VS UiMT - VS Tutorial
(BS +    Tutors)
2:00 - 3:30 JKV JKV BS UiMT - VS BS  
3:30 - 4:00 Tea/ coffee/ Snacks
4:00 - 6:00 Tutorial
(IS + Tutors)
(JKV + Tutors)
(JKV + Tutors)
(BS + Tutors)
(JKV + Tutors)


 Speakers  Unity in Mathematics Talks  Tutors
SRG - Sudhir R. Ghorpade
IIT Bombay
VS - V. Srinivas
TIFR, Mumbai
Mrinmoy Datta
ARP - Arunkumar R. Patil
SGGS, Nanded
TNV - T. N.  Venkataramana
TIFR, Mumbai   
 Joydip Saha
IS - Indranath Sengupta
IIT Gandhinagar
   Gaurab Tripathi
JKV -  Jugal  K. Verma
IIT Bombay
BS - Balwant Singh
CEBS, Mumbai


Topics  covered

SRG   Sudhi R Ghorpade (in Weeks 1 and 2)

  • Evolution of the theory of equations and a historical overview of Galois theory
  • Permutations, Cycles and transpositions, Symmetric Functions
  • Fundamental Theorem for Symmetric Polynomials, Newton’s identities
  • Polynomials and their roots, Fundamental Theorem of Algebra, Irreducibility criteria for polynomials over Q
  • Resultants and Discriminants
  •  Norm, Trace and Discriminant


ARP   Arunkumar R. Patil

  • Group actions, Cauchy’s theorem
  • Sylow’s theorems and Structure Theorem for finite abelian groups


IS Indranath Sengupta (in Weeks 1 and 2)

  • Simple groups: simplicity of An and PSL(2, p)
  • Solvable groups, Composition series and Jordan-Holder Theorem
  • Field extensions; basic properties
  • Splitting fields and normal extensions
  • Separable extensions, Primitive element theorem
  • Construction by ruler and compass, classical Greek problems (in Week 2)


JKV  J.K.Verma (in Week 2)

  • Galois groups and Galois extensions
  • Fundamental Theorem of Galois Theory
  • Applications to: Regular polygons, Fundamental theorem of algebra, Cubic equations
  • Calculating Galois groups of cubics and quartics
  • Solvable extensions and Galois’ criteria for solvability


BS  Balwant  Singh (in Week 2)

  • Finite fields
  • Construction of algebraic closure of a field, Examples
  • Roots of unity, cyclotomic extensions
  • Cyclic extensions, Hilbert’s Theorem 90, Artin-Schreier extensions


Unity in Mathematics Talks

  • VS    V.  Srinivas   :   Connection with fundamental groups
  • TNV   T.  N. Venkataramana :    Connection with monodromy groups


Message to the Participants of ISL on Galois Theory, Nanded

Galois theory is one of the most beautiful topics in mathematics. We have planned an exciting programme for you and we hope you will enjoy it and benefit from it. However, in order to get the most out of this course, we recommend that you follow the guidelines given below.

PRE-REQUISITES: Participants are expected to be familiar with Linear Algebra and rudiments  of Abstract Algebra (at least some basic working knowledge of groups, rings, fields and vector spaces). You may like to review this before coming to Nanded. There are several books available on Linear Algebra as well as Abstract Algebra, and also a vast amount of resources on the Internet. We recommend in particular, M. Artin’s Algebra. A low-priced Indian edition is readily available in many bookshops and on the Internet; see, for example:

Artin’s book covers both linear algebra and abstract algebra, and it will suffice to be familiar with Chapters 1-4 and 10 from this book. Another possibility is to look A First Course in Abstract Algebra, by J. B. Fraleigh, also available in Indian reprint at bookstores or Flipkart.

More enterprising persons among you may also be able to get a free soft copy of these books on the Internet by doing a Google search with keywords such as author, title, pdf.

Those who prefer watching and listening to lectures rather than read books may see at least the first few lectures of a nice video course on algebra at Harvard by Prof. Benedict Gross:

For the “Unity in Mathematics” talks, it will be desirable to have some exposure to basics of topology and complex analysis. Standard references here are Topology, by J. R. Munkres and Complex Analysis, by L. Ahlfors.

DURING THE INSTRUCTIONAL SCHOOL: There will be at least 3 hours of lectures and 2 hours of ttutorials almost every day. A lot of topics will get covered in a rather short time. It is important that you keep pace with it and not miss any lectures or tutorials. Also you are expected to put in at least 3 hours of self-study each day to go over the material covered on that day and to solve problems. Do not hesitate to ask questions during or after the lectures, and please be sure to take an active part in the tutorials. Do not plan any other extraneous activity or visit during the days of the school. You may, however, wish to take a day off on Sunday, 7th December 2014 and plan a touristic visit on your own to the city of Nanded or nearby places such as Ellora caves near Aurangabad. Outstation participants will be paid a daily allowance on this day instead of free breakfast, lunch and dinner. We wish you a pleasant and fruitful stay in Nanded.

- Sudhir Ghorpade and Arunkumar Patil

     Actual Participants


Sr. SID Full Name Gender


State Position Univ/ Inst.
Year of Passing M.Sc./ M.A Ph.D. Deg. Date
1 6726 Mr. Nitin Shridhar Darkunde Male SCHOOL OF MATHEMATICAL SCIENCES,
Maharashtra Assistant Professor UNIVERSITY OF PUNE, PUNE 2008  
2 6900 Mr Uday Subhash Divyaveer Male Swami Ramanand Teerth Marathwada University Maharashtra Assistant Professor


3 6862 Dr. Surendranath Reddy Male Swami Ramanand Teerth Marathwada University Maharashtra Assistnat Professor University of Hyderabad 2006 11 Jul 2012
4 6810 Mr Mahesh Sahebrao Wavare Male RAJARSHI SHAHU MAHAVIDYALAYA, LATUR Maharashtra Assistant Professor and Head PUNE UNIVERSITY, PUNE 2002  
5 6901 Mr. Sharad Bhagwanrao Kadam Male Anandibai Raorane Arts, Commerce & Science College, Vaibhavwadi Maharashtra Assistant Professor School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded 2012  
6 7084 Mr. Dipak Sandu Jadhav Male Smt. Chandibai Himathmal Mansukhani College Maharashtra Assistant Professor University of Mumbai 2008  
7 6779 Dr. Rupali Jain Female SRTM University Maharashtra Assistant Professor Dr. BAMU, Aurangabad 2005 21 Feb 2014
8 7387 Mr. Subhash Krishnan Male K. J. Somaiya College of Science and Commerce Maharashtra Associate Professor Mumbai University 1990  
9 7274 Mr. Amol Khandagale Male Department of Mathematics, Dr. Babasaeb Ambedkar Marathwada University, Aurangabad Maharashtra Asst Prof. M. Sc. 2006  
10 7214 Ms. Jita Parida Female Priyadarshini Indira Mahavidyalaya , Sambalpur University Orissa Lecturer Sambalpur University 2004  
11 6938 Mr. Bhairunath Shivaji Satpute Male Vaidyanath College,Parli-V. Maharashtra Assiatant Professor Shivaji Univesity 2002  
12 6758 Miss Syed Zarqua Mohammedi Female S.G.G.S IE&T Maharashtra lecturer S.R.T.M.University 2011  
13 7391 Mr Prashant Gurulingayya Swami Male SJCET, Palghar Maharashtra Assistant Professor S.R.T.M.University NANDED 2010  
14 7248 Ms. Sudeshna Roy Female IIT BOMBAY Maharashtra PhD JADAVPUR UNIVERSITY 2013  
15 7550 Mr. Samriddho Roy Male IIT Bombay Maharashtra PhD Jadavpur University 2013  
16 7422 Mr. Arindam Jana Male IIT BOMBAY Maharashtra Phd RAMAKRISHNA MISSION VIDYAMANDIRA 2012  
17 7423 Mr. Avijit Panja Male IIT BOMBAY Maharashtra Phd RAMAKRISHNA MISSION VIDYAMANDIRA 2012  
18 7094 Mr. Sachin Pandurang Basude Male School of Mathematical Sciences Maharashtra Ph.D student Swami Ramanand Teerth Marathwada University 2011  
19 7251 Mr. Gobinda Rakshit Male IIT Bombay Maharashtra Research Scholar IIT Kanpur 2011  
20 6738 Ms. Tripti Dahiya Female The maharaja sayaji rao university of Baroda Gujarat Msc student the maharaja sayaji rao university of Baroda Appeared/ Awaiting Result  
21 7215 Dr. Pasham Narasimha Swamy Male GITAM University Andhra Pradesh Assistant Professor Osmania University 2002  
22 7546 Mr Umanath S Male JAMAL MOHAMED COLLEGE Tamil Nadu M.Sc STUDENT JAMAL MOHAMED COLLEGE Appeared/ Awaiting Result  
23 6819 Ms. Jyoti Singh Siwach Female THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA, VADODARA Gujarat MSc STUDENT UNIVERSITY Appeared/ Awaiting Result  
24 6756 Mr. Amit Sharma Male PRATAP INSTITUE OF TECHNOLOGY AND SCIENCE Rajasthan Assistant Professor Maharshi Dayanand University, Rohtak 2006  
25 7326 Ms. Sharmishtha Sunil Deshmukh Female G.R. Patil College Maharashtra Assistant Professor. S.R.T.M.U. Nanded 2012  
26 7239 Mr Rajesh Digambarrao Sonkamble Male EKLVAYA RESIDENTIAL SCHOOL Maharashtra Post Graduate Teacher SRTMU NANDED 2006  
27   Raju A. Muneshwar   Assistant Professor     Science College,Nanded. 1997  


How to reach


The Instructional School for Lecturers on Galois Theory will be held at the SGGS Institute of Engineering and Technology, Nanded. Nanded is a large city in the Marathwada region of Maharashtra and is well connected to most places in India by train. For example, there are direct trains from Amritsar, Delhi, Mumbai, Nagpur, Pune, Bangalore, Hyderabad, and Aurangabad. When making a train booking, beware that the official name of the Nanded Railway station in the IRCTC website is H SAHIB NANDED (being an abbreviation for Hazur Sahib Nanded) and the codename is NED. Currently, there are no direct flights to Nanded. But it is possibly to fly to Aurangabad or Hyderabad and reach Nanded by road in about 5 hours.In Nanded, the SGGS Institute of Engineering and Technologyis popularly known as SGGS Engineering College. It is located at Vishnupuri and is about 10 kms away from the Nanded railway Station (=Hazur Sahib Nanded Station).  This station has two sides named `Gokul Nagar Side' and `Main Side'. The Main side is just beside Platform no. 1. Get off at the Main side (do not go to Gokul Nagar side), and you will see an AutoRickshaw stand. One can ask auto rickshaw driver to take you to SGGS Engineering College Vishnupuri. It will cost you Rs. 100 (maximum) for an individual ride. A shared auto rickshaw ride costs Rs. 25/- per person.Auto rickshaw's are available easily everywhere in Nanded.

When the exact accommodation arrangements of participants are known, we will update this later or inform the selected participants individually.