A Basic Course in Number Theory
A Basic Course in Number Theory. Instructor: Prof. Dipendra Prasad, Department of Mathematics, IIT Bombay. This course intends to develop the basics of number theory touching upon many essential points such as the prime number theorem, quadratic reciprocity laws, Gauss theorem on the classification of binary quadratic forms, Brahmagupta-Pell equations, to quote a few. This course will enable a student to learn more advanced topics in number theory.
(from nptel.ac.in)
| Lecture 01 - Integers |
| Lecture 02 - Divisibility and Primes |
| Lecture 03 - Infinitude of Primes |
| Lecture 04 - Division Algorithm and the GCD |
| Lecture 05 - Computing the GCD and Euclid's Lemma |
| Lecture 06 - Fundamental Theorem of Arithmetic |
| Lecture 07 - Stories around Primes |
| Lecture 08 - Winding up on Primes and Introducing Congruences |
| Lecture 09 - Basic Results in Congruences |
| Lecture 10 - Residue Classes Modulo N |
| Lecture 11 - Arithmetic Modulo N, Theory and Examples |
| Lecture 12 - Arithmetic Modulo N, More Examples |
| Lecture 13 - Solving Linear Polynomials Modulo N, Part I |
| Lecture 14 - Solving Linear Polynomials Modulo N, Part II |
| Lecture 15 - Solving Linear Polynomials Modulo N, Part III |
| Lecture 16 - Solving Linear Polynomials Modulo N, Part IV |
| Lecture 17 - Chinese Remainder Theorem, The Initial Cases |
| Lecture 18 - Chinese Remainder Theorem, The General Case and Examples |
| Lecture 19 - Chinese Remainder Theorem, More Examples |
| Lecture 20 - Using the CRT, Square Roots of 1 in Zn |
| Lecture 21 - Wilson's Theorem |
| Lecture 22 - Roots of Polynomials of Zp |
| Lecture 23 - Euler φ-Function, Part I |
| Lecture 24 - Euler φ-Function, Part II |
| Lecture 25 - Primitive Roots, Part I |
| Lecture 26 - Primitive Roots, Part II |
| Lecture 27 - Primitive Roots, Part III |
| Lecture 28 - Primitive Roots, Part IV |
| Lecture 29 - Structure of Un, Part I |
| Lecture 30 - Structure of Un, Part II |
| Lecture 31 - Quadratic Residues |
| Lecture 32 - The Legendre Symbol |
| Lecture 33 - Quadratic Reciprocity Law, Part I |
| Lecture 34 - Quadratic Reciprocity Law, Part II |
| Lecture 35 - Quadratic Reciprocity Law, Part III |
| Lecture 36 - Quadratic Reciprocity Law, Part IV |
| Lecture 37 - The Jacobi Symbol |
| Lecture 38 - Binary Quadratic Forms |
| Lecture 39 - Equivalence of Binary Quadratic Forms |
| Lecture 40 - Discriminant of a Binary Quadratic Form |
| Lecture 41 - Reduction Theory of Integral Binary Quadratic Forms |
| Lecture 42 - Reduced Forms up to Equivalence, Part I |
| Lecture 43 - Reduced Forms up to Equivalence, Part II |
| Lecture 44 - Reduced Forms up to Equivalence, Part III |
| Lecture 45 - Sums of Squares, Part I |
| Lecture 46 - Sums of Squares, Part II |
| Lecture 47 - Sums of Squares, Part III |
| Lecture 48 - Beyond Sums of Squares, Part I |
| Lecture 49 - Beyond Sums of Squares, Part II |
| Lecture 50 - Continued Fractions - Basic Results |
| Lecture 51 - Dirichlet's Approximation Theorem |
| Lecture 52 - Good Rational Approximations |
| Lecture 53 - Continued Fraction Expansion for Real Numbers, Part I |
| Lecture 54 - Continued Fraction Expansion for Real Numbers, Part II |
| Lecture 55 - Convergents Give Better Approximations |
| Lecture 56 - Convergents are the Best Approximations, Part I |
| Lecture 57 - Convergents are the Best Approximations, Part II |
| Lecture 58 - Quadratic Irrationals as Continued Fractions |
| Lecture 59 - Some Basics of Algebraic Number Theory |
| Lecture 60 - Units in Quadratic Fields: The Imaginary Case |
| Lecture 61 - Units in Quadratic Fields: The Real Case |
| Lecture 62 - Brahmagupta-Pell Equations |
| Lecture 63 - Tying Some Loose Ends |
| References |
A Basic Course in Number Theory
Instructor: Prof. Dipendra Prasad, Department of Mathematics, IIT Bombay. This course intends to develop the basics of number theory touching upon many essential points.
|