Course11110113763 lectures
A Basic Course in Number Theory
Prof. Shripad Garge · IIT Bombay
About
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.
INTENDED AUDIENCE : Students with basic knowledge of Mathematics can take this course.
PREREQUISITES : Knowledge of basic group theory and ring theory will be useful but it is not necessary.
Lectures
Week 1
- Lecture 1 : Integers
- Lecture 2 : Divisibility and primes
- Lecture 3 : Infinitude of primes
- Lecture 4 : Division algorithm and the GCD
- Lecture 5 : Computing the GCD and Euclid’s lemma
Week 2
- Lecture 6 : Fundamental theorem of arithmetic
- Lecture 7 : Stories around primes
- Lecture 8 : Winding up on `Primes' and introducing `Congruences'
- Lecture 9 : Basic results in congruences
- Lecture 10 : Residue classes modulo n
Week 3
- Lecture 11 : Arithmetic modulo n, theory and examples
- Lecture 12 : Arithmetic modulo n, more examples
- Lecture 13 : Solving linear polynomials modulo n - I
- Lecture 14 : Solving linear polynomials modulo n - II
- Lecture 15 : Solving linear polynomials modulo n - III
Week 4
- Lecture 16 : Solving linear polynomials modulo n - 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
Week 5
- Lecture 21 : Wilson's theorem
- Lecture 22 : Roots of polynomials over ℤp
- Lecture 23 : Euler 𝜑-function - I
- Lecture 24 : Euler 𝜑-function - II
- Lecture 25 : Primitive roots - I
Week 6
- Lecture 26 : Primitive roots - II
- Lecture 27 : Primitive roots - III
- Lecture 28 : Primitive roots - IV
- Lecture 29 : Structure of Un - I
- Lecture 30 : Structure of Un - II
Week 7
- Lecture 31 : Quadratic residues
- Lecture 32 : The Legendre symbol
- Lecture 33 : Quadratic reciprocity law - I
- Lecture 34 : Quadratic reciprocity law - II
- Lecture 35 : Quadratic reciprocity law - III
- Lecture 36 : Quadratic reciprocity law - IV
Week 8
- 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
Week 9
- Lecture 42 : Reduced forms up to equivalence - I
- Lecture 43 : Reduced forms up to equivalence - II
- Lecture 44 : Reduced forms up to equivalence - III
- Lecture 45 : Sums of squares - I
- Lecture 46 : Sums of squares - II
- Lecture 47 : Sums of squares - III
Week 10
- Lecture 48 : Beyond sums of squares - I
- Lecture 49 : Beyond sums of squares - II
- Lecture 50 : Continued fractions - basic results
- Lecture 51 : Dirichlet's approximation theorem
- Lecture 52 : Good rational approximations
Week 11
- Lecture 53 : Continued fraction expansion for real numbers - I
- Lecture 54 : Continued fraction expansion for real numbers - II
- Lecture 55 : Convergents give better approximations
- Lecture 56 : Convergents are the best approximations - I
- Lecture 57 : Convergents are the best approximations - II
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Bombay
- Department
- Mathematics
- Teacher
- Prof. Shripad Garge
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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