This
note in number theory explains standard topics in algebraic and analytic number
theory. Topics covered includes: Absolute values and discrete valuations,
Localization and Dedekind domains, ideal class groups, factorization of ideals,
Etale algebras, norm and trace, Ideal norms and the Dedekind-Kummer
thoerem, Galois extensions, Frobenius elements, Complete fields and valuation
rings, Local fields and Hensel's lemmas , Extensions of complete DVRs,
Totally ramified extensions and Krasner's lemma , Dirichlet's unit theorem,
Riemann's zeta function and the prime number theorem, The functional equation ,
Dirichlet L-functions and primes in arithmetic progressions, The analytic class
number formula, The Kronecker-Weber theorem, Class field theory, The main
theorems of global class field theory, Tate cohomology, profinite groups,
infinite Galois theory, Local class field theory, Global class field theory and
the Chebotarev density theorem.
This PDF covers the
following topics related to Number Theory : Divisibility, Prime Numbers, The
Linear Diophantine Equation , Congruences, Linear Congruences, The Chinese
Remainder Theorem, Public-Key Cryptography, Pseudoprimes, Polynomial
Congruences with Prime Moduli, Polynomial Congruences with Prime Power
Moduli, The Congruence, General Quadratic Congruences, The Legendre Symbol
and Gauss’ Lemma, Quadratic Reciprocity, Primitive Roots, Arithmetic
Functions, Sums of Squares, Pythagorean Triples, Fermat’s Last Theorem,
Continued Fractions, Simple Continued Fractions, Rational Approximations to
Irrational Numbers, Periodic Continued Fractions, Continued Fraction
Expansion, Pell’s Equation.
Analytic
number theory provides some powerful tools to study prime numbers, and most of
our current knowledge of primes has been obtained using these tools. Topics
covered includes: Primes and the Fundamental Theorem of Arithmetic, Arithmetic
functions: Elementary theory, Dirichlet series and Euler products and Asymptotic
estimates, Distribution of primes: Elementary results and Proof of the Prime
Number Theorem, Primes in arithmetic progressions.
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.
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.
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.