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
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.

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.