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.
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.
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.
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.
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 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.