Course11110111762 lectures
Galois Theory
Prof. Dilip Patil · IIT Bombay
About
Galois Theory is showpiece of a mathematical unification which brings together several differentbranches of the subject and creating a powerful machine for the study problems of considerablehistorical and mathematical importance. This course is an attempt to present the theory in such alight, and in a manner suitable for undergraduate and graduate students as well as researchers.This course will begin at the beginning. The quadratic formula for solving polynomials of degree2 has been known for centuries and is still an important part of mathematics education. Thecorresponding formulas for solving polynomials of degrees 3 and 4 are less familiar. Theseexpressions are more complicated than their quadratic counterpart, but the fact that they exist comesas no surprise. It is therefore altogether unexpected that no such formulas are available for solvingpolynomials of degree ≥ 5. A complete answer to this intriguing problem is provided by Galoistheory. In fact Galois theory was created precisely to address this and related questions aboutpolynomials. This feature might not be apparent from a survey of current textbooks on universitylevel algebra.This course develops Galois theory from historical perspective and I have taken opportunity to weavehistorical comments into lectures where appropriate. It provides a platform for the developmentof classical as well as modern core curriculum of Galois theory. Classical results by Abel, Gauss,Kronecker, Legrange, Ruffini and Galois are presented and motivation leading to a modern treatmentof Galois theory. The celebrated criterion due to Galois for the solvability of polynomials by radicals.The power of Galois theory as both a theoretical and computational tool is illustrated by a study ofthe solvability of polynomials of prime degree.The participant is expected to have a basic knowledge of linear algebra, but other that the course islargely self-contained. Most of what is needed from fields and elementary theory polynomials…
Lectures
Week 1
- Lecture 1 : Historical Perspectives
- Lecture 2 : Examples of Fields
- Lecture 3 : Polynomials and Basic properties
- Lecture 4 : Polynomial Rings
- Lecture 5 : Unit and Unit Groups
Week 2
- Lecture 6 : Division with remainder and prime factorization
- Lecture 7 : Zeroes of Polynomials
- Lecture 8 : Polynomial functions
- Lecture 9 : Algebraically closed Fields and statement of FTA
- Lecture 10 : Gauss’s Theorem(Uniqueness of factorization)
Week 3
- Lecture 11 : Digression on Rings homomorphism, Algebras
- Lecture 12 : Kernel of homomorphisms and ideals in K[X],Z
- Lecture 13 : Algebraic elements
- Lecture 14 : Examples
- Lecture 15 : Minimal Polynomials
Week 4
- Lecture 16 : Characterization of Algebraic elements
- Lecture 17 : Theorem of Kronecker
- Lecture 18 : Examples
- Lecture 19 : Digression on Groups
- Lecture 20 : Some examples and Characteristic of a Ring
Week 5
- Lecture 21 : Finite subGroups of the Unit Group of a Field
- Lecture 22 : Construction of Finite Fields
- Lecture 23 : Digression on Group action-I
- Lecture 24 : Automorphism Groups of a Field Extension
- Lecture 25 : Dedekind-Artin Theorem
- Lecture 26 : Galois Extension
Week 6
- Lecture 27 : Examples of Galois extension
- Lecture 28 : Examples of Automorphism Groups
- Lecture 29 : Digression on Linear Algebra
- Lecture 30 : Minimal and Characteristic Polynomials, Norms, Trace of elements
- Lecture 31 : Primitive Element Theorem for Galois Extension
Week 7
- Lecture 32 : Fundamental Theorem of Galois Theory
- Lecture 33 : Fundamental Theorem of Galois Theory(Contd)
- Lecture 34 : Cyclotomic extensions
- Lecture 35 : Cyclotomic Polynomials
- Lecture 36 : Irreducibility of Cyclotomic Polynomials over Q
- Lecture 37 : Reducibility of Cyclotomic Polynomials over Finite Fields
Week 8
- Lecture 38 : Galois Group of Cyclotomic Polynomials
- Lecture 39 : Extension over a fixed Field of a finite subGroup is Galois Extension
- Lecture 40 : Digression on Group action II
- Lecture 41 : Correspondence of Normal SubGroups and Galois sub-extensions
- Lecture 42 : Correspondence of Normal SubGroups and Galois sub-extensions(Contd)
Week 9
- Lecture 43 : Inverse Galois problem for Abelian Groups
- Lecture 44 : Elementary Symmetric Polynomials
- Lecture 45 : Fundamental Theorem on Symmetric Polynomials
- Lecture 46 : Gal (K[X1,X2,…,Xn]/K[S1,S2,...,Sn])
- Lecture 47 : Digression on Symmetric and Alternating Group
Week 10
- Lecture 48 : Discriminant of a Polynomial
- Lecture 49 : Zeroes and Embeddings
- Lecture 50 : Normal Extensions
- Lecture 51 : Existence of Algebraic Closure
- Lecture 52 : Uniqueness of Algebraic Closure
Week 11
- Lecture 53 : Proof of The Fundamental Theorem of Algebra
- Lecture 54 : Galois Group of a Polynomial
- Lecture 55 : Perfect Fields
- Lecture 56 : Embeddings
- Lecture 57 : Characterization of finite Separable extension
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Bombay
- Department
- Mathematics
- Teacher
- Prof. Dilip Patil
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
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