Chapter 2. Classical Number Theory

A 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...

41 items

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Chapter 2. Classical Number Theory

A 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...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Pythagorean Triples

A Pythagorean triple is a triple of positive integers

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Pell Equations

A Pell equation is a Diophantine equation of the form

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Sums of Squares

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Catalan-Type Equations

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Exponential Diophantine Equations

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Rational and Integral Points

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Geometry of Diophantine Problems

A 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.

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Squares Modulo $n$

A quadratic congruence is a congruence involving a square. The basic form is

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Legendre Symbol

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Jacobi Symbol

The Legendre symbol

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Euler Criterion

Euler 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 $p$....

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Quadratic Reciprocity

The theory of quadratic residues asks a fundamental question:

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Gauss Sums

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Higher Reciprocity Laws

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Computational Aspects

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Euclidean Algorithm Revisited

The 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.

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Finite Continued Fractions

A finite continued fraction is an expression of the form

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Infinite Continued Fractions

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Rational Approximations

Many important numbers are irrational:

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Convergents

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Pell Equations via Continued Fractions

Recall that a Pell equation has the form

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Diophantine Approximation

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Algebraic Integers

The ordinary integers

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Minimal Polynomials

An 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...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Number Fields

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Ring of Integers

In ordinary arithmetic, the integers

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Norm and Trace

Let $K$ be a number field of degree

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Unique Factorization Failure

One of the central properties of the ordinary integers is unique factorization.

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Ideals and Prime Ideals

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Class Groups

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Units and Dirichlet Unit Theorem

Let $K$ be a number field and let

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Discriminants

The 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.

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Principal Ideals

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

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Dedekind Domains

Ordinary integers satisfy several remarkable properties simultaneously:

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Valuations and Absolute Values

In ordinary analysis, the absolute value

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

$p$-Adic Numbers

The 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...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Local Fields

Classical number theory studies arithmetic globally over fields such as

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Ramification of Primes

In the ordinary integers, every nonzero integer factors uniquely into prime numbers.

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Decomposition and Inertia Groups

Let

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Frobenius Automorphisms

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