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
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)
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
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
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
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
Lecture 32 - Fundamental Theorem of Galois Theory
Lecture 33 - Fundamental Theorem of Galois Theory (Continued...)
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
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 (Continued...)
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
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
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
Lecture 58 - Primitive Element Theorem
Lecture 59 - Equivalence of Galois extensions and Normal-Separable extensions
Lecture 60 - Operation of Galois Group of Polynomial on the set of zeroes
Lecture 61 - Discriminants
Lecture 62 - Examples for further study
