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An extension of degree 2 of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760701.png" /> (cf. [[Extension of a field|Extension of a field]]). Any quadratic field has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760702.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760704.png" />, that is, it is obtained by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760705.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760706.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760707.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760708.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q0760709.png" />. Therefore any quadratic field has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607011.png" /> is a square-free integer that is uniquely determined by the field. In what follows, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607012.png" /> will always be taken to be this integer.
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{{TEX|done}}
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607014.png" /> is called a real, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607015.png" /> an imaginary, quadratic field.
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$$
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%\newcommand{\Q}{\mathbf{Q}}
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%\newcommand{\Z}{\mathbf{Z}}
 +
$$
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An extension of degree 2 of the field of rational numbers $\Q$ (cf. [[Extension of a field|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.
  
As a fundamental basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607016.png" />, that is, a basis of the ring of integers of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607017.png" /> over the ring of rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607018.png" />, one can take
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When $d>0$, $\Q\bigl(\sqrt d\bigr)$ is called a real, and when $d<0$ an imaginary, quadratic field.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607019.png" /></td> </tr></table>
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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
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607020.png" /></td> </tr></table>
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$$\left\{1,\sqrt d\right\} \quad \text{when}~d \equiv 2,3 \pmod 4$$
  
The [[Discriminant|discriminant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607022.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607023.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607025.png" />) and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607026.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607028.png" />).
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The [[Discriminant|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607029.png" />) with a finite unit group. This group has order 4 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607030.png" /> (and generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607031.png" />), order 6 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607032.png" /> (and generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607033.png" />), and order 2 (and generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607034.png" />) for all other imaginary quadratic fields.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607036.png" /> is the group of order 2 generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607038.png" /> is the infinite cyclic group generated by a fundamental unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607039.png" />. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607041.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607042.png" /> is a quadratic character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607044.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607046.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607047.png" />, then the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607048.png" /> is prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607049.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607050.png" />, and has two prime divisors when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607051.png" />.
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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:
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607052.png" /></td> </tr></table>
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$$ \frac{\ln{(hR)}}{\ln{\sqrt{|D|}}} \to 1 \quad \text{as}~ \left|D\right| \to \infty, $$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607055.png" /> are, respectively, the class number, the regulator (cf. [[Regulator of an algebraic number field|Regulator of an algebraic number field]]) and the discriminant of the field) shows that the class number tends to infinity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607056.png" />. There are exactly 9 imaginary quadratic fields of class number 1 (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607057.png" />, see [[#References|[2]]]). 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 [[#References|[3]]], [[#References|[4]]]). The analogous property for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607058.png" />-primary component of the class group follows from Gauss' theory of genera.
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where $h$, $R$ and $D$ are, respectively, the class number, the regulator (cf. [[Regulator of an algebraic number field|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 {{Cite|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 {{Cite|AnCh}}, {{Cite|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 [[#References|[5]]]) enables one to construct Abelian extensions of imaginary quadratic fields in an explicit form.
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The theory of complex multiplication (see {{Cite|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|Binary quadratic form]]).
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Many of the arithmetic properties of quadratic fields can be reformulated in terms of the theory of binary quadratic forms (cf. [[Binary quadratic form]]).
  
====References====
+
====Comments====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|1355542}} {{MR|1534414}} {{MR|0195803}} {{ZBL|0614.00005}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H.M. Stark, "A complete determination of the complex quadratic fields with class-number one" ''Michigan Math. J.'' , '''14''' (1967) pp. 1–27</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N.C. Ankeny, S. Chowla, "On the divisiblity of the class number of quadratic fields" ''Pacific J. Math.'' , '''5''' (1955) pp. 321–324</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Y. Yamamoto, "On umramified Galois extensions of quadratic number fields" ''Osaka J. Math.'' , '''7''' (1970) pp. 57–76</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) pp. Chapt. 13 {{MR|0215665}} {{ZBL|0153.07403}} </TD></TR></table>
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An effective version of the result $\lim_{D \to -\infty}h(D)=\infty$ has been proved by B.H. Gross and D.B. Zagier {{Cite|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====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|AnCh}}||valign="top"| N.C. Ankeny, S. Chowla, "On the divisiblity of the class number of quadratic fields" ''Pacific J. Math.'', '''5''' (1955) pp. 321–324
 +
|-
 +
|valign="top"|{{Ref|BoSh}}||valign="top"| Z.I. Borevich, I.R. Shafarevich, "Number theory", Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|1355542}} {{MR|1534414}} {{MR|0195803}} {{ZBL|0614.00005}}
 +
|-
 +
|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.) A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967) pp. Chapt. 13 {{MR|0215665}} {{ZBL|0153.07403}}
 +
|-
 +
|valign="top"|{{Ref|St}}||valign="top"| H.M. Stark, "A complete determination of the complex quadratic fields with class-number one" ''Michigan Math. J.'', '''14''' (1967) pp. 1–27
 +
|-
 +
|valign="top"|{{Ref|Ya}}||valign="top"| Y. Yamamoto, "On umramified Galois extensions of quadratic number fields" ''Osaka J. Math.'', '''7''' (1970) pp. 57–76
 +
|-
 +
|valign="top"|{{Ref|GrZa}}||valign="top"| B.H. Gross, D.B. Zagier, "Heegner points and derivatives of $L$-series" ''Invent. Math.'', '''84''' (1986) pp. 225–320 {{MR|833192}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"| Serge Lang, ''Cyclotomic fields I'' Springer (1978) {{ISBN|0-387-90307-0}}
 +
|-
 +
|}
  
====Comments====
+
[[Category:Number theory]]
An effective version of the result <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607059.png" /> has recently been proved by B.H. Gross and D.B. Zagier [[#References|[a1]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.H. Gross, D.B. Zagier, "Heegner points and derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076070/q07607060.png" />-series" ''Invent. Math.'' , '''84''' (1986) pp. 225–320 {{MR|833192}} {{ZBL|}} </TD></TR></table>
 

Latest revision as of 17:41, 12 November 2023


$$ %\newcommand{\Q}{\mathbf{Q}} %\newcommand{\Z}{\mathbf{Z}} $$ 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
How to Cite This Entry:
Quadratic field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_field&oldid=24118
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article