ISL Galois Theory (2014)

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.
Tutorials:
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

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 A_n 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:

http://www.flipkart.com/algebra-2-e-english-2nd/p/itmdx5gwcgrgbjxh?pid=9788120343290

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:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

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

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