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.

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.

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.

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.

This note covers the following topics: Algebraic numbers and algebraic
integers, Ideals, Ramification theory, Ideal class group and units, p-adic
numbers, Valuations, p-adic fields.