🔢Algebraic Number Theory Unit 10 – Quadratic and Cyclotomic Fields

Key Concepts and Definitions

• Algebraic number theory studies algebraic numbers, which are roots of polynomials with rational coefficients
• Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients
• Quadratic fields are number fields of degree 2 over the rational numbers $\mathbb{Q}$
  • Quadratic fields have the form $\mathbb{Q}(\sqrt{d})$, where $d$ is a square-free integer
• Cyclotomic fields are number fields generated by roots of unity
  • The $n$-th cyclotomic field is denoted by $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n$-th root of unity
• Ideals generalize the concept of divisibility in rings and play a crucial role in algebraic number theory
• Norm and trace are important functions in algebraic number theory that provide information about elements and ideals
• Discriminant is a value associated with a number field that measures its "complexity" and ramification

Quadratic Fields: Structure and Properties

• Quadratic fields are the simplest examples of algebraic number fields
• The discriminant of a quadratic field $\mathbb{Q}(\sqrt{d})$ is given by $\Delta = \begin{cases} d & \text{if } d \equiv 1 \pmod{4} \\ 4d & \text{if } d \equiv 2, 3 \pmod{4} \end{cases}$
• The ring of integers of a quadratic field depends on the discriminant
  • If $d \equiv 1 \pmod{4}$, then the ring of integers is $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$
  • If $d \equiv 2, 3 \pmod{4}$, then the ring of integers is $\mathbb{Z}[\sqrt{d}]$
• Units in quadratic fields have a specific structure described by Dirichlet's unit theorem
• The class number of a quadratic field measures the failure of unique factorization in its ring of integers
• Quadratic reciprocity law relates the solvability of quadratic congruences to the Legendre symbol

Cyclotomic Fields: Basics and Construction

• Cyclotomic fields are generated by roots of unity and have a rich algebraic structure
• The $n$-th cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of $\zeta_n$ over $\mathbb{Q}$
  • $\Phi_n(x) = \prod_{1 \leq k \leq n, \gcd(k,n)=1} (x - \zeta_n^k)$
• The degree of the $n$-th cyclotomic field is given by Euler's totient function $\varphi(n)$
• Cyclotomic fields have a unique factorization property, as their rings of integers are always principal ideal domains
• The Galois group of a cyclotomic field is isomorphic to the multiplicative group of units modulo $n$, denoted by $(\mathbb{Z}/n\mathbb{Z})^\times$
• Cyclotomic fields play a crucial role in the proof of Fermat's Last Theorem and the Kronecker-Weber theorem

Algebraic Integers in Quadratic and Cyclotomic Fields

• Algebraic integers in quadratic fields have the form $a + b\sqrt{d}$, where $a, b \in \mathbb{Z}$ or $\mathbb{Z}[\frac{1}{2}]$
• The norm of an algebraic integer $\alpha = a + b\sqrt{d}$ in a quadratic field is given by $N(\alpha) = a^2 - db^2$
• In cyclotomic fields, algebraic integers are linear combinations of powers of $\zeta_n$ with integer coefficients
  • $\alpha = a_0 + a_1\zeta_n + \cdots + a_{\varphi(n)-1}\zeta_n^{\varphi(n)-1}$, where $a_i \in \mathbb{Z}$
• The norm of an algebraic integer in a cyclotomic field is the product of its conjugates
• Algebraic integers in quadratic and cyclotomic fields form rings, which are integral domains
• The discriminant of the ring of integers in a quadratic or cyclotomic field is related to the field discriminant

Ideal Theory and Factorization

• Ideals in rings of integers generalize the concept of divisibility and factorization
• Principal ideals are generated by a single element, while non-principal ideals require multiple generators
• In quadratic fields, the ideal class group measures the failure of unique factorization
  • The class number is the order of the ideal class group
• Cyclotomic fields have a unique factorization property, as their rings of integers are principal ideal domains
• The splitting of primes in quadratic and cyclotomic fields is governed by the Dedekind-Kummer theorem
  • The theorem relates the splitting behavior to congruence conditions on the prime
• The ideal norm is a multiplicative function that extends the notion of the absolute value to ideals
• Factorization of ideals in quadratic and cyclotomic fields can be computed using algorithms such as the Buchmann-Lenstra algorithm

Galois Theory Applications

• Galois theory studies the symmetries of algebraic extensions and provides a powerful framework for understanding their structure
• The Galois group of a quadratic field $\mathbb{Q}(\sqrt{d})$ is either trivial or isomorphic to $\mathbb{Z}/2\mathbb{Z}$
  • The Galois group is trivial if and only if $d$ is a perfect square
• The Galois group of the $n$-th cyclotomic field is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$, the multiplicative group of units modulo $n$
• Galois theory can be used to prove the irreducibility of cyclotomic polynomials over $\mathbb{Q}$
• The Kronecker-Weber theorem states that every abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field
  • This theorem establishes a deep connection between abelian extensions and cyclotomic fields
• Galois theory provides a framework for understanding the splitting behavior of primes in extensions and the structure of ideal class groups

Computational Techniques and Examples

• Computing the ring of integers in quadratic fields involves finding an integral basis
  • For $d \equiv 1 \pmod{4}$, the integral basis is $\{1, \frac{1+\sqrt{d}}{2}\}$
  • For $d \equiv 2, 3 \pmod{4}$, the integral basis is $\{1, \sqrt{d}\}$
• Computing the class number of a quadratic field can be done using the Minkowski bound and the Hurwitz class number formula
• Computing the Galois group of a cyclotomic field involves determining the structure of $(\mathbb{Z}/n\mathbb{Z})^\times$
• Factoring ideals in quadratic and cyclotomic fields can be done using the Buchmann-Lenstra algorithm or the Shor-Weinberger algorithm
• Computing the discriminant of a quadratic or cyclotomic field involves evaluating the discriminant formula
• Solving Diophantine equations, such as Pell's equation, can be done using techniques from algebraic number theory

Real-World Applications and Advanced Topics

• Algebraic number theory has applications in cryptography, such as the construction of secure cryptographic schemes
  • The hardness of certain problems in algebraic number theory, such as the shortest vector problem, is used in lattice-based cryptography
• Algebraic number theory is used in the study of Diophantine equations, which have connections to various areas of mathematics
• The Langlands program, a vast generalization of class field theory, seeks to unify various branches of mathematics through the study of automorphic forms and Galois representations
• Elliptic curves, which are used in cryptography and have applications in factoring integers, are closely related to algebraic number theory
• The study of algebraic number theory has led to the development of powerful computational tools, such as the LLL algorithm and the PARI/GP computer algebra system
• Algebraic number theory has connections to other areas of mathematics, such as algebraic geometry, representation theory, and analytic number theory