MA2317: Introduction to Number Theory

Michaelmas term, 2011-12

Lecturer: Masha Vlasenko


We begin with the classical questions such as factorization, distribution of primes and finding integer solutions to algebraic equations. Second part of the course will deal with number fields and basics of algebraic number theory. Our ultimate goal will be to understand the quadratic reciprocity law and Dirichlet's theorem on arithmetic progressions in the context of class field theory. The last theme of the course is counting points on algebraic varieties over finite fields and the Weil conjectures. Though the statement of these conjectures is quite elementary, their proof took decades and had very high influence on the development of contemporary number theory.

The prerequisites are basic analysis, linear algebra and group theory. Lectures will be accompanied by numerous exercises intended to practice calculations with algebraic numbers, p-adic numbers and finite fields. There will be also bi-weekly tutorials in the form of problem-solving sessions.

In case one is motivated for further studies, we could recommend to look through the book of David A. Cox for higher reciprocity laws and to learn about the Birch and Swinnerton-Dyer conjecture. The latter concerns with the L-function of an elliptic curve which will be one of our last themes.

Exam paper with solutions(published on May 24)

Syllabus

  1. Euclidean algorithm and Unique Factorization
  2. Calculations with Residue Classes and the Quadratic Reciprocity Law
  3. Some Modern Problems of Elementary Number Theory: Primality Tests, Factorization and Public Key Cryptosystems
  4. Distribution of Primes, Riemann Hypothesis, Primes in Arithmetic Progressions


  5. Polynomials and Algebraic Extensions; Algebraic Numbers
  6. Diophantine Equations of Degree Two and Class Numbers
  7. Irrationality and Diophantine Approximation
  8. p-adic numbers
  9. Cubic Diophantine Equations and Elliptic curves
  10. Finite fields
  11. Zeta function of an algebraic variety and Weil conjectures

Literature

[IrelandRosen]
K.Ireland, M.Rosen, A Classical Introduction to Modern Number Theory
[Koblitz1]
N. Koblitz, A course in Number Theory and Cryptography
[Serre]
J.-P. Serre, A Course in Arithmetic
[PanchishkinManin]
Yu.I.Manin, A.A.Panchishkin, Introduction to Number Theory, Part I: Problems and Tricks
[Lang1]
S. Lang, Math Talks for Undergraduates
[Lang2]
S. Lang, Algebra
[Davenport]
H.Davenport, The Higher arithmetic
[Cox]
D.A. Cox, Primes of the form x^2 + n y^2
[HardyWright]
G.H.Hardy, E.M.Wright, An Introduction to the Theory of Numbers
[Koblitz2]
N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions
[Chandrasekharan]
K. Chandrasekharan, Introduction to Analytic Number Theory


Miscellaneous


PARI/GP calculator