TEW Galois Theory (2019)

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Venue: Bharathiar University,  Coimbatore, Tamil Nadu
Date:  8th, Jul 2019 to 13th, Jul 2019

 

School Convener(s)

Name Prof.  J.K. Verma Dr. R. Rakkiyappan
Mailing Address

Department of Mathematics,
IIT Bombay
Mumbai 400076.

Assistant Professor,
Department of Mathematics, BharathiarUniversity
Coimbatore 641 046

 


 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 p-gon 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, Orbit-Stabilizer 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): (Dedekind-Artin) # 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 08-07-2019 AVJ   SS   AVJ   AVJ+SM+PS
Tuesday 09-07-2019 SS   AVJ   SS   SS+SM+PS
Wednesday 10-07-2019 MK   DPP   MK   MK+DPP+PS
Thursday 11-07-2019 DPP   MK   DPP   DPP+MK+PS
Friday 12-07-2019 JKV   TC   RB   JKV+DPP+PS
Saturday 13-07-2019 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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