Learn · Introduction to Algebraic Structures
Lecture 52 of 58Week 11 Ring of reside classes and its group of units, polynomial ring over a field, primitive roots and linear…
Polynomial ring over a field, division algorithm and consequences.
Prof. Mahesh Kakde and Prof. R. Venkatesh · IISc Bangalore
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- Introduction to Algebraic Structures, IISc Bangalore (at NPTEL)
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All the lectures
- Introduction to mathematics and set theory
- Set identities, Cartesian product, relations and functions
- Logic and construction of new sets
- Functions
- Properties of functions
- Composing functions and Cantor's theorem
- Don Zaiger's proof: Primes of the form 4k+1 can written as sum of two squares.
- Axiom of choice and Paritition of a set.
- Correspondence between partitions of a set and equivalence relations on it.
- Partially ordered sets, well-ordering principle and Zorn's lemma.
- Defining natural numbers: Peano axioms.
- Order properties of natural numbers and notion of cardinality of a set.
- Cardinality of a finite set.
- Countable sets
- Uncountable sets
- Schroder-Bernstein theorem
- Integers and Rational Numbers
- The Construction of Rational Numbers and Counting Principles
- Counting in Two Ways
- Binomial Coefficients and Properties
- The Pigeonhole Principle
- Formal Series
- Formal Series and Applications
- Solving Linear Recurrences
- Principle of Inclusion and Exclusion
- Möbius Function on Finite Posets
- Möbius Inversion Formula
- Möbius Inversion Formula and Applications
- Introduction to Graph Theory, Graphs and Special Graph Families
- Graph Representations and Basic Graph Operations
- Handshake Lemma and Graph Connectivity
- Cycles, Bipartite Graphs, and Trees
- Trees and Their Properties
- Eulerian and Hamiltonian Graphs
- Matchings and Graph Coloring
- Number theory and divisibility
- Greatest Common Divisor - I
- Greatest Common Divisor - II
- LCM, primes and prime factorisation
- Consequences of prime factorisation and infinitude of primes
- Euler's proof of infinitude of primes
- Congruencies and modular arithmetic
- Diophantine equations and residue classes
- Solving Diophantine equations in one variable and Chinese remainder theorem
- Solving Diophantine equations using CRT
- Proof of Hensel's lemma
- Introduction to Euler's totient function
- More on totient function
- Mobius inversion formula
- Euler's theorem
- Ring structure on set of residue classes
- Polynomial ring over a field, division algorithm and consequences.
- Wilson's theorem, Lagrange's theorem and introduction to cyclic groups
- Classification of cyclic unit groups modulo m and proof.
- Primitive roots modulo m and application in solving linear congruences
- More on solving congruence equations and introduction to quadratic residue and Legendre symbol
- Computing Legendre symbol: Euler's criterion and Law of quadratic reciprocity
- Dirichlet's theorem on primes in AP: Proof of special cases