TEW Galois Theory (2019)
Venue:  Bharathiar University, Coimbatore, Tamil Nadu 
Date:  8th, Jul 2019 to 13th, Jul 2019 
Name  Prof. J.K. Verma  Dr. R. Rakkiyappan 
Mailing Address 
Department of Mathematics, 
Assistant Professor, 
Speakers with their affiliations:
Name of the speaker 
Affiliation 
A. V. Jayanthan  IIT Madras 
Sarang Sane  IIT Madras 
Sarang Sane  CMI, Chennai 
Dilip Patil  IISc, Bangalore 
T. Tamizh Selvam  M. S. Univ., Tiruneveli 
J. K. Verma  IIT Bombay 
R. Balasubramanian  IIT Bombay 
List of tutors:
Name of the tutor 
Affiliation / official postal address 
Dr. Shreedevi Masuti 
IIT Dharwad 
Dr. Parangama Sarkar 
CMI, Chennai. 
Syllabus:
The topics covered covered by Speakers
Name of the speaker 
Affiliation 
Topic (sections in M. Artin) 
A. V. Jayanthan  IIT Madras  Examples of fields (15.2) Algebraic extensions (15.3) Ruler Compass constructions (15.5) 
Sarang Sane  IIT Madras  Multiple roots (15.6) Finite fields (15.7) Primitive element theorem (15.8) 
Sarang Sane  CMI, Chennai  Symmetric functions (16.1, 16.2) Splitting field of a polynomial (16.3) Isomorphisms of splitting fields (16.4) 
Dilip Patil  IISc, Bangalore  Fixed fields and Artin’s Theorem (16.5) Galois extensions and their examples (16.6) Group actions and their examples 
T. Tamizh Selvam  M. S. Univ., Tiruneveli  Fundamental Theorem of Galois Theory (16.7) Illustrative examples (16.8) 
J. K. Verma  IIT Bombay  Cyclotomic extensions (16.10) Cyclic extensions (16.11) 
R. Balasubramanian  IIT Bombay  Introduction to algebraic number fields Chebotarev Density Theorem 

Lectures by A. V. Jayanthan, IIT Madras

First lecture: Introduced the notion of a field and discussed many examples of fields and their extensions, degree of field extensions and examples.

Second lecture: Proved the multiplicative property of degree of field extensions. Defined algbraic elements and algebraic extensions and gave examples. Proved that finite implies algebraic and gave an example to show that the converse is not true.

Third lecture: Sketched the proof of the transitivity of algebraicity of field extensions. Introduced the Greek questions in Euclidean geometry and ruler and compass constructions. Proved that if a real number is constructible, then it is contained in an extension of Q of degree 2 r for some r ≥ 1. Proved that if a regular pgon is constructible, then p = 2 r + 1 for some r > 0.

Lectures by Sarang Sane, IIT Madras

First Lecture: I discussed splitting fields and defined normal extensions. We studied how the Euclidean algorithm and related concepts like gcd, etc. don’t change under a field extension. We used this to study when a polynomial has multiple roots in characteristic 0 and p > 0. We defined separable extensions and saw that in characteristic 0, all extensions are separable. There were running examples through the lecture.

Second Lecture: We discussed finite fields, particularly that there is a unique (upto isomorphism) finite field of order q = p r for each prime p > 0 and positive integer r and it occurs as the splitting field of X q − X, which we then denoted by F q . Along the way we proved that for a finite field F, its group of units F ∗ is cyclic.

Third Lecture: We finished studying finite fields by considering when F p r ⊆ F p s and analyzing the decomposition of X q − X into its irreducible factors. We stated the primitive element theorem and proved it in characteristic 0.

Lectures by Manoj Kummini, CMI, Chennai
 Three lectures were given. The first lecture dealt with symmetric polynomials, with applications to Galois theory. In the second lecture, splitting fields of polynomials were discussed. In the third lecture, the uniqueness of splitting fields (up to isomorphisms) was proved, and examples of automorphism groups of finite extensions were calculated.

Lectures by Dilip Patil, IISc, Bangalore

Lecture 1: Group operations, Examples, OrbitStabilizer Theorem, Symmetric Polynomials, Automorphism groups, Galois group of a field extension L  K and its operation on L and the zero set V L ( f ) of a polynomial f ∈ K [ X ] .

Lecture 2: For a finite field extension, L  K, the inequality (without proof): (DedekindArtin) # Gal ( L  K ) ≤ [ L : K ] . Definition and examples of Galois extensions. Some (easy) computation of Galois groups. Description of the Galois group of a simple Galois extension K ( x ) K in terms of the zeros of μ x , K (the minimal polynomial of x over K).

Lecture 3: Proved Artin’s Theorem: G ⊆ Aut L finite subgroup. Then L  Fix G L is a finite Galois extension with Galois group G. Further, proved characterization (equivalent formulations) of Galois extension. Discussed some examples.

Lectures by Tamizh Chelvam, M. S. University, Tiruneveli

First lecture: the Fundamental Theorem for Galois Theory was proved. After proving the same, some illustrative examples were given in which the number of intermediate fields is obtained by imposing some condition on the Galois group Gal ( K  F ) .

Second lecture: Galois theory for cubic equations was discussed in which the Galois group for a cubic polynomial is realised as A 3 or S 3 according to the nature of its discriminant. Using this, some problems were solved where the Galois group of certain cubic polynomials were computed.

Lectures by J. K. Verma

First Lecture: I discussed cyclotomic extensions. It was proved that the Galois group of Q ( ζ n ) /Q ) is isomorphic to the group U ( n ) of units of Z/nZ and hence its order is φ ( n ) . The irreducibility of the n th cyclotomic polynomial Φ n ( x ) over Q was also proved. A recursive formula for Φ n ( x ) was derived.

Second Lecture: Primitive elements of intermediate subfields of Q ( ζ p ) were determined using a cyclic generator of the multiplicative group F × p . Using this knowledge, Gauss’ criterion for constructibility of a regular polygon of n sides was proved. It was proved that there are infinitely many primes of the form p ≡ 1 ( mod n ) using the cyclotomic polynomials.

Lectures by R. Balasubramanian

First lecture: was a quick introduction to algebraic number theory. Number fields and their rings of integers were introduced. The notion of a Dedekind domain was discussed. It was proved that O K is a Dedekind domain for a number field K. Norm and traces of elements of K were introduced via embedding of K in C. Ramification of primes was discussed and the e f g = n theorem was explained.

Second lecture: Quadratic number fields were introduced. The ramification of primes in this context was discussed. Their relation of ramification √ of primes with discriminant of the number field was discussed. Ramification of primes in Q ( d ) in terms of the Legendre symbol was mentioned. The decomposition and the inertia groups were introduced. This leads to the definition of the Frobenius element. The main result of these two lectures was the Chebotorov Density Theorem. This result implies that there are infinitely primes of the form p ≡ a ( mod n ) where ( a, n ) = 1 This is deduced by a simple application of the Chebotorov Density Theorem for the cyclotomic extension.
Time Table
Day  Date  Lecture 1 9.30 to 11.00 
Tea 11.00 to 11.15 
Lecture 2 11.15 to 12.45 
Lunch 12.45 to 2.00 
Lecture 3 2:00 to 3.30 
Tea 3.30 to 3.45 
Tutorial 3.45 to 5.15 
Monday  08072019  AVJ  SS  AVJ  AVJ+SM+PS  
Tuesday  09072019  SS  AVJ  SS  SS+SM+PS  
Wednesday  10072019  MK  DPP  MK  MK+DPP+PS  
Thursday  11072019  DPP  MK  DPP  DPP+MK+PS  
Friday  12072019  JKV  TC  RB  JKV+DPP+PS  
Saturday  13072019  JKV  TC  RB  TC+DPP+PS 
Full forms for the abbreviations of speakers and tutors:

Dr. AV Jayanthan (AVJ)

Prof. Dilip Patil (DPP)

Prof. R. Balasubramanian (RB)

Dr. Sarang Sane (SS)

J.K. Verma (JKV)

Dr. Manoj Kummini (MK)

T. Tamizh Chelvam (TC)

Dr. Shreedevi Masuti (SM)

Dr. Parangama Sarkar (PS)
List of actual Participants
Sr  SID  Full Name  Gender  Affiliation  Position in College/ University  University/ Institute M.Sc./ M.A.  Year of Passing M.Sc./ M.A  Ph.D. Deg. Date 
1  27571  Mr. Vivek Dabra  Male  Thapar Institute of Engineering & Technology  PhD  Thapar Institute of Engineering & Technology (Master in Engineering)  2016  20/07/2016 
2  27916  Dr. K Pattabiraman  Male  Government Arts College(Autonomous)  Asst. Prof.  Annamalai University  2004  
3  27942  Dr. Swaminathan A  Male  Government Arts College(Autonomous)  Asst. Prof.  Bharathidasan University  2005  04/10/2013 
4  28546  Mr Vignesh Perumal  Male  Patrician college of Arts and Science  Asst. Prof.  Manomaniam sundaranar  2009  
5  28622  Mr. K Dhurai  Male  Government college of Engineering, Dharmapuri.  Assistant professor  Bharathidasan university  2005  
6  28637  Dr Ramasamy Ct  Male  Alagappa Government Arts College  Asst. Prof.  Alagappa University  2006  28/01/2012 
7  28669  Ms. Kaviya V L  Female  Ramanujan Institute For Advanced Study In Mathematics,University Of Madras  MSc Student  Ramanujan Institute For Advanced Study In Mathematics, University Of Madras  Appeared / Awaiting Result  
8  28682  Mrs Ilakkiya R  Female  working at Nehru Institute of Technology  Asst. Prof.  Gandhigram Rural University  2010  
9  28686  Mr. Sanjit Das  Male  VIT Chennai  VIT Chennai; School of Advanced Sciences;  IIT Kharagpur  2005  15/09/2012 
10  28699  Mr Saurabh Rana  Male  The LNM Institute of Information Technology  PhD Scholar  CCS university, Meerut  2013  
11  28725  Mrs R Santhakumari  Female  Sri Ramakrishna College Of Arts And Science  Asst. Prof  Nirmala College For Women  
12  28798  Ms S.Vinnarasi Vincy  Female  St.joseph's college for women  Assistant professor  M.sc  2016  
13  28807  Ms. Ekta Bindal  Female  IIT(ISM) Dhanbad  PHD  Maharshi Dayanand University,Rohtak  2016  
14  28855  Mrs Suganya Baskaran  Female  St. Joseph's college for women  Asst. Prof.  Bharatiyar university  2014  01/07/2018 
15    Ms. Sharmila.V  Female  Department of Mathematics, BU  Research Scholar       
16    Ms. Abi.M  Female  Department of Mathematics, BU  PG Student       
17    Ms. Anitha. V  Female  Department of Mathematics, BU  PG Student       
18    Ms. Arivazhagan. J  Female  Department of Mathematics, BU  PG Student       
19    Ms. Chithra. S  Female  Department of Mathematics, BU  PG Student       
20    Ms. Deepa.M  Female  Department of Mathematics, BU  PG Student       
21    Ms. Devi Poornima. P  Female  Department of Mathematics, BU  PG Student       
22    Ms. Divya.J  Female  Department of Mathematics, BU  PG Student       
23    Ms. Fathimuthu Johra. A  Female  Department of Mathematics, BU  PG Student       
24    Ms. Gifteena Hingis. Y.M  Female  Department of Mathematics, BU  PG Student       
25    Ms. Gomathi. D  Female  Department of Mathematics, BU  PG Student       
26    Ms. Indhu. G  Female  Department of Mathematics, BU  PG Student       
27    Ms. Infanta Anu Josy.R  Female  Department of Mathematics, BU  PG Student       
28    Ms. Jamilaa Afreen. M  Female  Department of Mathematics, BU  PG Student       
29    Ms. Janani. K  Female  Department of Mathematics, BU  PG Student       
30    Ms. Jayashree. M  Female  Department of Mathematics, BU  PG Student       
31    Ms. Kalaivani. B  Female  Department of Mathematics, BU  PG Student       
32    Ms. Kalpana. M  Female  Department of Mathematics, BU  PG Student       
33    Ms. Kannika. M  Female  Department of Mathematics, BU  PG Student       
34    Mr. Karthik. S  Male  Department of Mathematics, BU  PG Student       
35    Ms. Keerthana. N  Female  Department of Mathematics, BU  PG Student       
36    Ms. Kousalya. K  Female  Department of Mathematics, BU  PG Student       
37    Mr. Logarasu. S  Male  Department of Mathematics, BU  PG Student       
38    Ms. Mohanapriya. K  Female  Department of Mathematics, BU  PG Student       
39    Ms. Monika Sri. S  Female  Department of Mathematics, BU  PG Student       
40    Ms. Mownicka. S  Female  Department of Mathematics, BU  PG Student       
41    Ms. Mythili. S  Female  Department of Mathematics, BU  PG Student       
42    Ms. Neepha. M.V  Female  Department of Mathematics, BU  PG Student       
43    Mr. Parthiban.R  Male  Department of Mathematics, BU  PG Student       
44    Mr. Praveen. A  Male  Department of Mathematics, BU  PG Student       
45    Ms. Praveena. R.R  Female  Department of Mathematics, BU  PG Student       
46    Ms. Poorani. A.K.  Female  Department of Mathematics, BU  PG Student       
47    Ms. Rasikaa. S  Female  Department of Mathematics, BU  PG Student       
48    Ms. Reena. K  Female  Department of Mathematics, BU  PG Student       
49    Ms. Renuka. T  Female  Department of Mathematics, BU  PG Student       
50    Ms. Saidhivya. A  Female  Department of Mathematics, BU  PG Student       
51    Ms. Saranya. B  Female  Department of Mathematics, BU  PG Student       
52    Ms. Shanmugapriya. K  Female  Department of Mathematics, BU  PG Student       
53    Ms. Shrisoundarya. B  Female  Department of Mathematics, BU  PG Student       
54    Ms. Sivaranjani. M  Female  Department of Mathematics, BU  PG Student       
55    Ms. Suvitha. K  Female  Department of Mathematics, BU  PG Student       
56    Ms. Vaishnavi.P  Female  Department of Mathematics, BU  PG Student       
57    Ms. Varshni. M  Female  Department of Mathematics, BU  PG Student       
58    Ms. Yugavathi. V  Female  Department of Mathematics, BU  PG Student       
59    Ms. Yuvasri. N.K.  Female  Department of Mathematics, BU  PG Student       
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