Quadratic field

$$ %\newcommand{\Q}{\mathbf{Q}} %\newcommand{\Z}{\mathbf{Z}} $$ An extension of degree 2 of the field of rational numbers $\Q$ (cf. Extension of a field). Any quadratic field has the form $\Q\bigl(\sqrt d\bigr)$, where $d \in \Q$, $\sqrt d \notin \Q$, that is, it is obtained by adjoining $\sqrt d$ to $\Q$. $\Q\bigl(\sqrt{d_1}\bigr)=\Q\bigl(\sqrt{d_2}\bigr)$ if and only if $d_1=c^2d_2$, where $c \in \Q$. Therefore any quadratic field has the form $\Q\bigl(\sqrt d\bigr)$, where $d$ is a square-free integer that is uniquely determined by the field. In what follows, $d$ will always be taken to be this integer.

When $d>0$, $\Q\bigl(\sqrt d\bigr)$ is called a real, and when $d<0$ an imaginary, quadratic field.

As a fundamental basis of $\Q\bigl(\sqrt d\bigr)$, that is, a basis of the ring of integers of the field $\Q\bigl(\sqrt d\bigr)$ over the ring of rational integers $\Z$, one can take

$$\left\{1,\frac{1+\sqrt d}{2}\right\} \quad \text{when}~d \equiv 1 \pmod 4$$

and

$$\left\{1,\sqrt d\right\} \quad \text{when}~d \equiv 2,3 \pmod 4$$

The discriminant $D$ of $\Q\bigl(\sqrt d\bigr)$ is equal to $d$ when $d \equiv 1 \pmod 4$ and to $4d$ when $d \equiv 2,3 \pmod 4$.

Imaginary quadratic fields are the only type (apart from $\Q$) with a finite unit group. This group has order 4 for $\Q\bigl(\sqrt{-1}\bigr)$ (and generator $\sqrt{-1}$), order 6 for $\Q\bigl(\sqrt{-3}\bigr)$ (and generator $\bigl(1+\sqrt{-3}\bigr)/2$), and order 2 (and generator $-1$) for all other imaginary quadratic fields.

For real quadratic fields the unit group is isomorphic to the direct product $\{\pm 1\} \times \{\epsilon\}$ , where $\{\pm 1\}$ is the group of order 2 generated by $-1$ and $\{\epsilon\}$ is the infinite cyclic group generated by a fundamental unit $\epsilon$. For example, for $\Q\bigl(\sqrt 2\bigr)$, $\epsilon = 1+\sqrt 2$.

The factorization rule for rational prime divisors in a quadratic field has a simple formulation: Associated with $\Q\bigl(\sqrt d\bigr)$ is a quadratic character $\chi$ on $\Z$ modulo $\left|D\right|$. If $p$ is a prime number and $(D,p)=1$, then the divisor $(p)$ is prime in $\Q\bigl(\sqrt d\bigr)$ when $\chi(p)=-1$, and has two prime divisors when $\chi(p)=1$.

The divisor class group of quadratic fields has been studied more extensively than that of any other class of fields. For imaginary quadratic fields, the Brauer–Siegel theorem (stating that for algebraic number fields of fixed degree the following asymptotic formula holds:

$$ \frac{\ln{(hR)}}{\ln{\sqrt{|D|}}} \to 1 \quad \text{as}~ \left|D\right| \to \infty, $$

where $h$, $R$ and $D$ are, respectively, the class number, the regulator (cf. Regulator of an algebraic number field) and the discriminant of the field) shows that the class number tends to infinity as $d \to -\infty$. There are exactly 9 imaginary quadratic fields of class number 1 (for $d=-1,-2,-3,-7,-11,-19,-43,-67,-163$, see [St]). For real quadratic fields it is not known (1990) whether there is an infinite number of fields of class number 1. There are an infinite number of (both real and imaginary) quadratic fields whose class number is divisible by a given natural number (see [AnCh], [Ya]). The analogous property for the $2$-primary component of the class group follows from Gauss' theory of genera.

The theory of complex multiplication (see [CaFr]) enables one to construct Abelian extensions of imaginary quadratic fields in an explicit form.

Many of the arithmetic properties of quadratic fields can be reformulated in terms of the theory of binary quadratic forms (cf. Binary quadratic form).

Comments

An effective version of the result $\lim_{D \to -\infty}h(D)=\infty$ has been proved by B.H. Gross and D.B. Zagier [GrZa].

A quadratic field is an Abelian extension of $\mathbb{Q}$, and if the discriminant is $D$ then the conductor (cf Conductor of an Abelian extension) is $(D)$. This is a special case of the Fuhrerdiskriminantenproduktformel.

References

[AnCh]	N.C. Ankeny, S. Chowla, "On the divisiblity of the class number of quadratic fields" Pacific J. Math., 5 (1955) pp. 321–324
[BoSh]	Z.I. Borevich, I.R. Shafarevich, "Number theory", Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966) MR1355542 MR1534414 MR0195803 Zbl 0614.00005
[CaFr]	J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) pp. Chapt. 13 MR0215665 Zbl 0153.07403
[St]	H.M. Stark, "A complete determination of the complex quadratic fields with class-number one" Michigan Math. J., 14 (1967) pp. 1–27
[Ya]	Y. Yamamoto, "On umramified Galois extensions of quadratic number fields" Osaka J. Math., 7 (1970) pp. 57–76
[GrZa]	B.H. Gross, D.B. Zagier, "Heegner points and derivatives of $L$-series" Invent. Math., 84 (1986) pp. 225–320 MR833192
[La]	Serge Lang, Cyclotomic fields I Springer (1978) ISBN 0-387-90307-0