10.1 Quadratic fields and their properties
4 min read•july 30, 2024
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 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
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- 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
- 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 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
- 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
Key Terms to Review (17)
Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They play a crucial role in number theory, particularly in the study of unique factorization and properties of number fields, connecting various concepts like integral bases and prime ideals.
Class group computation: Class group computation refers to the process of determining the structure and elements of the class group associated with a number field, particularly in the context of quadratic fields. This involves identifying the ideals of the ring of integers that represent unique factorization and understanding how these ideals relate to one another, ultimately revealing information about the arithmetic properties of the field. The class group plays a crucial role in determining whether the ring of integers in a number field is a unique factorization domain.
Class number: The class number is an important invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It reflects how many distinct ideal classes exist, indicating whether every ideal can be expressed uniquely as a product of prime ideals. A class number of one means that unique factorization holds, while a higher class number suggests complications in the structure of ideals within the number field.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Degree of extension: The degree of extension refers to the dimension of a field extension as a vector space over its base field. It measures how many elements from the larger field can be expressed as linear combinations of elements from the smaller field. Understanding the degree of extension is crucial for exploring properties like algebraic closures, norms, traces, and various structures in number theory.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Factorization in number fields: Factorization in number fields refers to the process of expressing an element of a number field as a product of irreducible elements or primes within that field. This concept is crucial as it extends the classical notion of factorization from integers to more complex structures, allowing for a richer understanding of arithmetic properties. Factorization plays an important role in the study of the norm and trace, as it helps to analyze how these concepts interact with ideals and their decomposition in different number fields.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial equation, formed by the automorphisms of a field extension that fix the base field. This concept helps us understand how different roots relate to one another and provides a powerful framework for analyzing the solvability of polynomials and the structure of number fields.
Hasse's Principle: Hasse's Principle is a concept in number theory that states if a certain property holds for a number in local fields (like the completions of the rational numbers at various primes), then it also holds for that number in the global field, specifically the rational numbers. This principle is crucial when analyzing whether certain equations have solutions in the rational numbers based on their behavior in local fields, connecting various properties in algebraic number theory, such as discriminants, integral bases, and quadratic fields.
Imaginary quadratic field: An imaginary quadratic field is a type of number field formed by adjoining the square root of a negative integer to the rational numbers. These fields are characterized by their unique properties related to the integers they contain, as well as their influence on algebraic number theory, particularly in the study of class groups and ideal class numbers.
Joseph Liouville: Joseph Liouville was a French mathematician known for his significant contributions to algebraic number theory, particularly through the introduction of Liouville's theorem and the study of algebraic integers. His work laid important groundwork in understanding properties of quadratic fields and their structures, influencing how mathematicians approached number theory and algebra.
Law of Quadratic Reciprocity: The law of quadratic reciprocity is a fundamental theorem in number theory that describes the conditions under which a quadratic equation has solutions modulo prime numbers. It provides a deep connection between the solvability of two different quadratic congruences, essentially allowing mathematicians to determine whether one prime can be expressed as a quadratic residue modulo another. This law is a cornerstone in understanding the properties of quadratic fields and their extensions, influencing how we approach other number theoretic concepts.
Quadratic field: A quadratic field is a type of number field that can be expressed in the form $$ ext{Q}(\sqrt{d})$$, where $$d$$ is a square-free integer. This concept is crucial for understanding the structure of algebraic integers and their properties, as well as examining how these fields relate to Galois theory and integral bases. Quadratic fields provide a rich framework for exploring topics like unit groups and the behavior of primes in number fields, linking various aspects of algebraic number theory together.
Rational Integers: Rational integers are the set of whole numbers that include both positive and negative numbers as well as zero. They are integral to the study of number theory, particularly in understanding the structure and properties of various number systems, including quadratic fields. Rational integers are essential for defining elements in fields formed by adjoining roots of polynomial equations.
Real quadratic field: A real quadratic field is a type of number field generated by adjoining the square root of a positive integer to the rational numbers, specifically of the form $$\mathbb{Q}(\sqrt{d})$$ where $$d$$ is a positive square-free integer. This concept connects to various important properties of algebraic integers, the structure of units, and the behavior of class groups within number theory.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Units in the field: Units in the field refer to elements within a field that have a multiplicative inverse, meaning that for a unit 'u', there exists another element 'v' such that 'u * v = 1'. In the context of quadratic fields, understanding units is essential as they help characterize the field's structure and relate to number theory concepts like unique factorization and the distribution of prime ideals.