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.
Author(s): Andrew Sutherland
This note covers the following topics: Primes in Arithmetic Progressions, Infinite products, Partial summation and Dirichlet series, Dirichlet characters, L(1, x) and class numbers, The distribution of the primes, The prime number theorem, The functional equation, The prime number theorem for Arithmetic Progressions, Siegel’s Theorem, The Polya-Vinogradov Inequality, Sums of three primes, The Large Sieve, Bombieri’s Theorem.
Author(s): Andreas Strombergsson
This note covers the following topics: Formal Power Series, Theta-functions, Analytic theory of partitions, Representation by squares.
Author(s): H. Rademacher
This lecture note is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.
Author(s): Prof. Abhinav Kumar
This note covers the following topics: Divisibility and Primes, Congruences, Congruences with a Prime-Power Modulus, Euler's Function and RSA Cryptosystem, Units Modulo an Integer, Quadratic Residues and Quadratic Forms, Sum of Powers, Fractions and Pell's Equation, Arithmetic Functions, The Riemann Zeta Function and Dirichlet L-Function.
Author(s): Dr. Anupam Saikia, NTPEL
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.
Author(s): A.J. Hildebrand
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.
Author(s): Wissam Raji
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.
Author(s): William Stein
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.
Author(s): Paul Garrett
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.
Author(s): R. D. Carmichael
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.
Author(s): Frederique Oggier
This note covers the following topics: Fermat’s Last Theorem , Euclid's Classification of Pythagorean Triples and Germain's General Approach.
Author(s): Reinhard Laubenbacher and David Pengelley
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.
Author(s): Peter J. Cameron
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