TEW Galois Theory (2019)

 Speakers with their affiliations:

List of tutors:

 Syllabus:  

The topics covered covered by Speakers

• 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

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

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