Lecture 6 of 62Week 2
Lecture 6 : Division with remainder and prime factorization
Prof. Dilip Patil · IIT Bombay
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- Galois Theory, IIT Bombay (at NPTEL)
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All the lectures
- 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(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
- 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)
- 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