NOC:Galois Theory


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