This note contains the
following subtopics: Classfield theory, homological formulation, harmonic
polynomial multiples of Gaussians, Fourier transform, Fourier inversion on
archimedean and p-adic completions, commutative algebra: integral extensions
and algebraic integers, factorization of some Dedekind zeta functions into
Dirichlet L-functions, meromorphic continuation and functional equation of zeta,
Poisson summation and functional equation of theta, integral representation of
zeta in terms of theta.
The
notes contain a useful introduction to important topics that need to be
addressed in a course in number theory. Proofs of basic theorems are presented
in an interesting and comprehensive way that can be read and understood even by
non-majors with the exception in the last three chapters where a background in
analysis, measure theory and abstract algebra is required.
This is a
textbook about classical elementary number theory and elliptic curves. The first
part discusses elementary topics such as primes, factorization, continued
fractions, and quadratic forms, in the context of cryptography, computation, and
deep open research problems. The second part is about elliptic curves, their
applications to algorithmic problems, and their connections with problems in
number theory.
Robert Daniel Carmichael (March
1, 1879 – May 2, 1967) was a leading American mathematician.The purpose of this
little book is to give the reader a convenient introduction to the theory of
numbers, one of the most extensive and most elegant disciplines in the whole
body of mathematics. The arrangement of the material is as follows: The five
chapters are devoted to the development of those elements which are essential to
any study of the subject. The sixth and last chapter is intended to give the
reader some indication of the direction of further study with a brief account of
the nature of the material in each of the topics suggested.
This
note explains the following topics:
Algebraic numbers, Finite continued fractions, Infinite continued fractions,
Periodic continued fractions, Lagrange and Pell, Euler’s totient function,
Quadratic residues and non-residues, Sums of squares and Quadratic forms.