Quadratic fields, extensions of rational numbers, form the foundation of algebraic number theory. They come in two flavors: real and imaginary, each with unique properties. These fields help us understand more complex number systems and their behavior.

The ring of integers in quadratic fields reveals fascinating structures. Real fields have infinite unit groups, while imaginary ones are finite. This difference impacts factorization, leading to the study of class groups and numbers, key concepts in number theory.

Quadratic Fields and Properties

Definition and Basic Structure

• Quadratic field represents a number field of degree 2 over rational numbers Q, denoted as Q(√d) where d stands for a square-free integer
• Elements in Q(√d) take the form a + b√d, with a and b being rational numbers
• Serve as fundamental examples of algebraic number fields, crucial for understanding more complex number theory concepts
• Conjugate of a + b√d in Q(√d) defined as a - b√d
• Norm of an element calculated as (a + b√d)(a - b√d) = a^2 - db^2
• Discriminant of Q(√d) equals d if d ≡ 1 (mod 4), and 4d otherwise, determining various field properties

Types and Characteristics

• Quadratic fields categorized as real (d > 0) or imaginary (d < 0)
• Real quadratic fields (Q(√2), Q(√3)) exhibit distinct algebraic and arithmetic properties from imaginary fields (Q(√-1), Q(√-3))
• Real quadratic fields contain infinitely many units, while imaginary quadratic fields have finite unit groups
• Imaginary quadratic fields form a discrete subring of the complex plane
• Real quadratic fields create a lattice structure in the real plane

Classification by Discriminants

Discriminant Properties

• Discriminant of Q(√d) determines crucial arithmetic properties and classifies these fields
• Positive discriminant indicates real quadratic fields (d > 0)
• Negative discriminant signifies imaginary quadratic fields (d < 0)
• Discriminant parity (odd or even) influences the ring of integers structure and integral basis
• Size and factorization of discriminant affect class number and other arithmetic invariants

Types of Discriminants

• Real quadratic fields (d > 0) have positive discriminants, either ≡ 1 (mod 4) or ≡ 0 (mod 4)
• Imaginary quadratic fields (d < 0) possess negative discriminants, either ≡ 1 (mod 4) or ≡ 0 (mod 4)
• Fundamental discriminants (square-free and ≡ 1 mod 4, or square-free and divisible by 4) exhibit simpler arithmetic properties
• Examples of fundamental discriminants include 5 (for Q(√5)) and -4 (for Q(√-1))
• Non-fundamental discriminants, like 12 (for Q(√3)), have more complex properties

Integral Basis and Ring of Integers

Integral Basis Structure

• Ring of integers OQ(√d) comprises elements that are roots of monic polynomials with integer coefficients
• For d ≡ 2 or 3 (mod 4), integral basis takes form {1, √d}
• When d ≡ 1 (mod 4), integral basis becomes {1, (1 + √d)/2}
• General integer form for d ≡ 2 or 3 (mod 4) a + b√d, with a and b as rational integers
• For d ≡ 1 (mod 4), general integer form a + b(1 + √d)/2, where a and b are rational integers
• Examples include Q(√2) with basis {1, √2} and Q(√5) with basis {1, (1 + √5)/2}

Properties of the Ring of Integers

• OQ(√d) forms a Dedekind domain, crucial for understanding ideal factorization
• Imaginary quadratic fields have rings of integers as discrete subrings of the complex plane
• Real quadratic fields form lattices in the real plane
• Norm of an algebraic integer in OQ(√d) always yields a rational integer
• Ring of integers determines the arithmetic properties of the quadratic field
• Unique factorization may fail in the ring of integers, leading to the study of ideal class groups

Unit and Class Groups

Unit Group Structure

• Unit group consists of invertible elements in the ring of integers
• Imaginary quadratic fields have finite unit groups, primarily roots of unity
• Only nine imaginary quadratic fields possess non-trivial units beyond ±1 (Q(√-1), Q(√-2), Q(√-3), Q(√-7), Q(√-11), Q(√-19), Q(√-43), Q(√-67), Q(√-163))
• Real quadratic fields contain infinite unit groups of form {±ε^n : n ∈ Z}, ε being the fundamental unit
• Fundamental unit found through continued fraction expansion of √d
• Regulator, defined as the natural logarithm of the fundamental unit, serves as an important invariant

Class Group and Number

• Class group measures the failure of unique factorization in the ring of integers
• Finite for all quadratic fields, with its order defined as the class number
• Class number equals 1 if and only if the ring of integers forms a unique factorization domain
• Class number formula connects class number, regulator, and other field invariants
• Examples of class number 1 fields include Q(√-1), Q(√-2), Q(√-3), Q(√-7), Q(√-11), Q(√-19), Q(√-43), Q(√-67), Q(√-163)
• Gauss's class number problem, concerning the finiteness of imaginary quadratic fields with a given class number, remains a significant area of research