Chapter 4. Algebraic Number Theory

A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...

35 items

Section	Title
1	Chapter 4. Algebraic Number Theory
2	Splitting Fields
3	Galois Groups
4	Finite Fields
5	Cyclotomic Fields
6	Ramification
7	Absolute Values
8	$p$-Adic Numbers
9	Completion of Fields
10	Hensel’s Lemma
11	Local-Global Principles
12	Adeles and Ideles
13	Abelian Extensions
14	Reciprocity Maps
15	Hilbert Class Fields
16	Local Class Field Theory
17	Global Class Field Theory
18	Modular Groups
19	Modular Functions
20	Modular Forms
21	Eisenstein Series
22	Cusp Forms
23	Hecke Operators
24	Modular Curves
25	Elliptic Curves and Modularity
26	The Modularity Theorem
27	Automorphic Forms
28	Automorphic Representations
29	The Langlands Program
30	Galois Representations
31	Functoriality
32	Automorphic $L$-Functions
33	Trace Formulas
34	Shimura Varieties
35	Geometric Langlands Theory

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Chapter 4. Algebraic Number Theory

A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Splitting Fields

A central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Galois Groups

A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Finite Fields

The familiar fields

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Cyclotomic Fields

One of the most important classes of number fields arises from the solutions of the equation

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Ramification

One of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Absolute Values

The ordinary absolute value on the real numbers measures magnitude:

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

$p$-Adic Numbers

The real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and...

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Completion of Fields

The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Hensel’s Lemma

One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Local-Global Principles

A central problem in number theory is determining whether an equation possesses rational or integral solutions.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Adeles and Ideles

The rational numbers may be studied through their completions:

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Abelian Extensions

A central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Reciprocity Maps

One of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Hilbert Class Fields

One of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Local Class Field Theory

Global class field theory studies finite abelian extensions of number fields such as

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Global Class Field Theory

One of the central goals of algebraic number theory is to classify field extensions of a number field

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Modular Groups

Modular forms begin with the action of certain matrix groups on the complex upper half-plane.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Modular Functions

The modular group acts on the upper half-plane by fractional linear transformations:

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Modular Forms

Modular forms are among the central objects of modern number theory.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Eisenstein Series

Among all modular forms, Eisenstein series are the most explicit and computationally accessible.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Cusp Forms

Modular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Hecke Operators

Modular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Modular Curves

The modular group acts on the upper half-plane by fractional linear transformations:

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Elliptic Curves and Modularity

An elliptic curve is simultaneously:

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

The Modularity Theorem

For centuries, elliptic curves and modular forms were studied as separate objects.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Automorphic Forms

Modular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Automorphic Representations

Classical modular form theory begins with analytic functions satisfying symmetry conditions.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

The Langlands Program

The Langlands program is one of the most ambitious and influential theories in modern mathematics.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Galois Representations

Galois groups encode the symmetries of algebraic equations and field extensions.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Functoriality

The Langlands program predicts that many different arithmetic objects are connected by systematic transfers.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Automorphic $L$-Functions

The Riemann zeta function

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Trace Formulas

Fourier analysis decomposes functions into harmonic frequencies.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Shimura Varieties

Modular curves parameterize elliptic curves and connect modular forms with arithmetic geometry.

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Geometric Langlands Theory

The classical Langlands program relates: