Quadratic fields
Introduction
A quadratic field is a type of number field generated by the square root of a number. More formally, a quadratic field is a field extension of the rational numbers \( \mathbb{Q} \) of degree 2. Quadratic fields are among the simplest types of number fields and have been extensively studied in algebraic number theory.
Definition and Basic Properties
A quadratic field \( \mathbb{Q}(\sqrt{d}) \) is defined as \( \mathbb{Q}(\sqrt{d}) = \{ a + b\sqrt{d} \mid a, b \in \mathbb{Q} \} \), where \( d \) is a square-free integer. The field \( \mathbb{Q}(\sqrt{d}) \) is a two-dimensional vector space over \( \mathbb{Q} \), with basis \( \{1, \sqrt{d}\} \).
Discriminant
The discriminant \( \Delta \) of a quadratic field \( \mathbb{Q}(\sqrt{d}) \) is given by: \[ \Delta = \begin{cases} d & \text{if } d \equiv 1 \pmod{4}, \\ 4d & \text{if } d \equiv 2 \text{ or } 3 \pmod{4}. \end{cases} \] The discriminant plays a crucial role in the arithmetic of quadratic fields, influencing the structure of the ring of integers and the behavior of primes.
Ring of Integers
The ring of integers \( \mathcal{O}_{\mathbb{Q}(\sqrt{d})} \) in a quadratic field \( \mathbb{Q}(\sqrt{d}) \) is the set of all algebraic integers in \( \mathbb{Q}(\sqrt{d}) \). For a quadratic field \( \mathbb{Q}(\sqrt{d}) \), the ring of integers is given by: \[ \mathcal{O}_{\mathbb{Q}(\sqrt{d})} = \begin{cases} \mathbb{Z}[\sqrt{d}] & \text{if } d \equiv 2 \text{ or } 3 \pmod{4}, \\ \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right] & \text{if } d \equiv 1 \pmod{4}. \end{cases} \]
Class Number
The class number \( h(d) \) of a quadratic field \( \mathbb{Q}(\sqrt{d}) \) is a fundamental invariant that measures the failure of unique factorization in the ring of integers \( \mathcal{O}_{\mathbb{Q}(\sqrt{d})} \). It is defined as the order of the ideal class group, which is the group of fractional ideals modulo principal ideals.
Class Number Formula
For a quadratic field \( \mathbb{Q}(\sqrt{d}) \), the class number can be computed using analytic methods involving the Dedekind zeta function of the field. The class number formula relates the residue of the Dedekind zeta function at \( s = 1 \) to the class number, the regulator, and the number of roots of unity in the field.
Units in Quadratic Fields
The units in the ring of integers \( \mathcal{O}_{\mathbb{Q}(\sqrt{d})} \) form a group under multiplication. For a quadratic field \( \mathbb{Q}(\sqrt{d}) \), the structure of the unit group depends on whether \( d \) is positive or negative.
Real Quadratic Fields
If \( d > 0 \), \( \mathbb{Q}(\sqrt{d}) \) is a real quadratic field. The unit group is infinite and is generated by a fundamental unit \( \epsilon \) and \(-1\). The fundamental unit can be found using the continued fraction expansion of \( \sqrt{d} \).
Imaginary Quadratic Fields
If \( d < 0 \), \( \mathbb{Q}(\sqrt{d}) \) is an imaginary quadratic field. The unit group is finite and consists of roots of unity. Specifically, the unit group is \( \{\pm 1\} \) for most \( d \), but for \( d = -1, -3 \), the unit groups are \( \{\pm 1, \pm i\} \) and \( \{\pm 1, \pm \omega\} \) respectively, where \( \omega = \frac{-1 + \sqrt{-3}}{2} \).
Galois Theory
Quadratic fields are Galois extensions of \( \mathbb{Q} \) with Galois group isomorphic to \( \mathbb{Z}/2\mathbb{Z} \). The non-trivial automorphism in the Galois group is the map \( \sigma \) defined by \( \sigma(\sqrt{d}) = -\sqrt{d} \).
Applications
Quadratic fields have applications in various areas of mathematics, including Diophantine equations, cryptography, and the theory of modular forms. They also play a role in the study of elliptic curves and L-functions.
Examples
Example 1: \( \mathbb{Q}(\sqrt{2}) \)
The field \( \mathbb{Q}(\sqrt{2}) \) is a real quadratic field with discriminant \( \Delta = 8 \). The ring of integers is \( \mathbb{Z}[\sqrt{2}] \), and the fundamental unit is \( 1 + \sqrt{2} \).
Example 2: \( \mathbb{Q}(\sqrt{-1}) \)
The field \( \mathbb{Q}(\sqrt{-1}) \) is an imaginary quadratic field with discriminant \( \Delta = -4 \). The ring of integers is \( \mathbb{Z}[i] \), and the unit group is \( \{\pm 1, \pm i\} \).
See Also
References
- Neukirch, Jürgen. Algebraic Number Theory. Springer-Verlag, 1999.
- Lang, Serge. Algebraic Number Theory. Addison-Wesley, 1970.
- Marcus, Daniel A. Number Fields. Springer-Verlag, 1977.