Chapter 2. Classical Number Theory – brain

Section	Title
1	Chapter 2. Classical Number Theory
2	Pythagorean Triples
3	Pell Equations
4	Sums of Squares
5	Catalan-Type Equations
6	Exponential Diophantine Equations
7	Rational and Integral Points
8	Geometry of Diophantine Problems
9	Squares Modulo $n$
10	Legendre Symbol
11	Jacobi Symbol
12	Euler Criterion
13	Quadratic Reciprocity
14	Gauss Sums
15	Higher Reciprocity Laws
16	Computational Aspects
17	Euclidean Algorithm Revisited
18	Finite Continued Fractions
19	Infinite Continued Fractions
20	Rational Approximations
21	Convergents
22	Pell Equations via Continued Fractions
23	Diophantine Approximation
24	Algebraic Integers
25	Minimal Polynomials
26	Number Fields
27	Ring of Integers
28	Norm and Trace
29	Unique Factorization Failure
30	Ideals and Prime Ideals
31	Class Groups
32	Units and Dirichlet Unit Theorem
33	Discriminants
34	Principal Ideals
35	Dedekind Domains
36	Valuations and Absolute Values
37	$p$ -Adic Numbers
38	Local Fields
39	Ramification of Primes
40	Decomposition and Inertia Groups
41	Frobenius Automorphisms

Chapter 2. Classical Number TheoryA Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless...

Pythagorean TriplesA Pythagorean triple is a triple of positive integers

Pell EquationsA Pell equation is a Diophantine equation of the form

Sums of SquaresOne of the oldest questions in number theory asks which integers can be written as sums of squares. Typical examples are

Catalan-Type EquationsA Catalan-type equation is a Diophantine equation involving powers whose values differ by a small amount. The classical example is

Exponential Diophantine EquationsAn exponential Diophantine equation is a Diophantine equation in which one or more unknowns appear as exponents. Typical examples include

Rational and Integral PointsA central problem in number theory is to study solutions of polynomial equations whose coordinates belong to a specified number system. Two important cases are:

Geometry of Diophantine ProblemsA Diophantine equation is first an arithmetic object. It asks for solutions in integers or rational numbers. But every polynomial equation also defines a geometric object.

Squares Modulo $n$A quadratic congruence is a congruence involving a square. The basic form is

Legendre SymbolLet $p$ be an odd prime and let $a\in\mathbb{Z}$. The Legendre symbol is defined by

Jacobi SymbolThe Legendre symbol

Euler CriterionEuler criterion gives an efficient way to decide whether an integer is a square modulo an odd prime. Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$....

Quadratic ReciprocityThe theory of quadratic residues asks a fundamental question:

Gauss SumsGauss sums arise from combining multiplicative and additive structures modulo a prime. They form one of the fundamental tools of analytic and algebraic number theory.

Higher Reciprocity LawsQuadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers.

Computational AspectsQuadratic residue theory is not only a theoretical subject. It also plays a major role in computational number theory, cryptography, primality testing, and algorithm design.

Euclidean Algorithm RevisitedThe Euclidean algorithm is one of the oldest and most important algorithms in mathematics. It computes the greatest common divisor of two integers using repeated division.

Finite Continued FractionsA finite continued fraction is an expression of the form

Infinite Continued FractionsFinite continued fractions correspond exactly to rational numbers. When the Euclidean algorithm never terminates, the continued fraction becomes infinite.

Rational ApproximationsMany important numbers are irrational:

ConvergentsThe convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.

Pell Equations via Continued FractionsRecall that a Pell equation has the form

Diophantine ApproximationDiophantine approximation studies how closely real numbers can be approximated by rational numbers.

Algebraic IntegersThe ordinary integers

Minimal PolynomialsAn algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a...

Number FieldsA number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying

Ring of IntegersIn ordinary arithmetic, the integers

Norm and TraceLet $K$ be a number field of degree

Unique Factorization FailureOne of the central properties of the ordinary integers is unique factorization.

Ideals and Prime IdealsIn ordinary integers, every number factors uniquely into primes. In many rings of algebraic integers, this property fails.

Class GroupsIn ordinary integers, every ideal is generated by a single element:

Units and Dirichlet Unit TheoremLet $K$ be a number field and let

DiscriminantsThe discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic.

Principal IdealsLet $R$ be a commutative ring. An ideal $I\subseteq R$ is called principal if there exists an element $\alpha\in R$ such that

Dedekind DomainsOrdinary integers satisfy several remarkable properties simultaneously:

Valuations and Absolute ValuesIn ordinary analysis, the absolute value

$p$-Adic NumbersThe real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic...

Local FieldsClassical number theory studies arithmetic globally over fields such as

Ramification of PrimesIn the ordinary integers, every nonzero integer factors uniquely into prime numbers.

Decomposition and Inertia GroupsLet

Frobenius AutomorphismsOne of the deepest ideas in algebraic number theory is that prime numbers possess hidden symmetry inside field extensions.