Difference between revisions of "Quadratic field"
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| − | An extension of degree 2 of the field of rational numbers | + | An extension of degree 2 of the field of rational numbers $\mathbf Q$ (cf. [[Extension of a field|Extension of a field]]). Any quadratic field has the form $\bf Q$$(\sqrt d)$, where $d \in \bf Q$, $\sqrt d \notin \bf Q$, that is, it is obtained by adjoining $\sqrt d$ to $\bf Q$. $\bf Q$$(\sqrt d_1)=\bf Q$$(\sqrt d_2)$ if and only if $d_1=c^2d_2$, where $c \in \bf Q$. Therefore any quadratic field has the form $\bf Q$$(\sqrt d)$, 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 | + | When $d>0$, $\bf Q$$(\sqrt d)$ is called a real, and when $d<0$ an imaginary, quadratic field. |
| − | As a fundamental basis of | + | As a fundamental basis of $\it Q$$(\sqrt d)$, that is, a basis of the ring of integers of the field $\bf Q$$(\sqrt d)$ over the ring of rational integers $\bf Z$, one can take |
| − | + | $$\left\{1,\frac{1+\sqrt d}{2}\right\} \quad \text{when}~d \equiv 1 \pmod 4$$ | |
and | and | ||
| − | + | $$\left\{1,\sqrt d\right\} \quad \text{when}~d \equiv 2,3 \pmod 4$$ | |
| − | The [[Discriminant|discriminant]] | + | The [[Discriminant|discriminant]] $D$ of $\bf Q$$(\sqrt d)$ 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 | + | Imaginary quadratic fields are the only type (apart from $\bf Q$) with a finite unit group. This group has order 4 for $\bf Q$$(\sqrt{-1})$ (and generator $\sqrt{-1}$), order 6 for $\bf Q$$(\sqrt{-3})$ (and generator $(1+\sqrt{-3})/2$), 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 | + | 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 $\bf Q$$(\sqrt 2)$, $\epsilon = 1+\sqrt 2$. |
| − | The factorization rule for rational prime divisors in a quadratic field has a simple formulation: Associated with | + | The factorization rule for rational prime divisors in a quadratic field has a simple formulation: Associated with $\bf Q$$(\sqrt d)$ is a quadratic character $\chi$ on $\bf Z$ modulo $|D|$. If $p$ is a prime number and $(D,p)=1$, then the divisor $(p)$ is prime in $\bf Q$$(\sqrt d)$ when $\chi(p)=-1$, and has two prime divisors when $\chi(p)=1$. |
The divisor class group of quadratic fields has been studied more extensively than that of any other class of fields. For imaginary quadratic fields, the Brauer–Siegel theorem (stating that for algebraic number fields of fixed degree the following asymptotic formula holds: | The divisor class group of quadratic fields has been studied more extensively than that of any other class of fields. For imaginary quadratic fields, the Brauer–Siegel theorem (stating that for algebraic number fields of fixed degree the following asymptotic formula holds: | ||
| − | + | $$ \frac{\ln{(hR)}}{\ln{\sqrt{|D|}}} \to 1 \quad \text{as}~ |D| \to \infty, $$ | |
| − | where | + | 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 [[#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 $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. | 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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====Comments==== | ====Comments==== | ||
| − | An effective version of the result | + | An effective version of the result $\lim_{D \to -\infty}h(D)=\infty$ has recently been proved by B.H. Gross and D.B. Zagier [[#References|[a1]]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.H. Gross, D.B. Zagier, "Heegner points and derivatives of | + | <table><TR><TD valign="top">[a1]</TD> <TD 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|}} </TD></TR></table> |
Revision as of 23:10, 22 April 2012
An extension of degree 2 of the field of rational numbers $\mathbf Q$ (cf. Extension of a field). Any quadratic field has the form $\bf Q'"`UNIQ-MathJax1-QINU`"'(\sqrt d_1)=\bf Q'"`UNIQ-MathJax2-QINU`"'(\sqrt d)$, 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$, $\bf Q'"`UNIQ-MathJax3-QINU`"'(\sqrt d)$, that is, a basis of the ring of integers of the field $\bf Q'"`UNIQ-MathJax4-QINU`"'\left\{1,\frac{1+\sqrt d}{2}\right\} \quad \text{when}~d \equiv 1 \pmod 4'"`UNIQ-MathJax5-QINU`"'\left\{1,\sqrt d\right\} \quad \text{when}~d \equiv 2,3 \pmod 4'"`UNIQ-MathJax6-QINU`"'(\sqrt d)$ 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 $\bf Q$) with a finite unit group. This group has order 4 for $\bf Q'"`UNIQ-MathJax7-QINU`"'(\sqrt{-3})$ (and generator $(1+\sqrt{-3})/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 $\bf Q'"`UNIQ-MathJax8-QINU`"'(\sqrt d)$ is a quadratic character $\chi$ on $\bf Z$ modulo $|D|$. If $p$ is a prime number and $(D,p)=1$, then the divisor $(p)$ is prime in $\bf Q'"`UNIQ-MathJax9-QINU`"' \frac{\ln{(hR)}}{\ln{\sqrt{|D|}}} \to 1 \quad \text{as}~ |D| \to \infty, $$ 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 [[#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 $2$-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. 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]]). ===='"`UNIQ--h-0--QINU`"'References==== <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) [https://mathscinet.ams.org/mathscinet/article?mr=1355542 MR1355542] [https://mathscinet.ams.org/mathscinet/article?mr=1534414 MR1534414] [https://mathscinet.ams.org/mathscinet/article?mr=0195803 MR0195803] [https://zbmath.org/?q=an%3A0614.00005 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 [https://mathscinet.ams.org/mathscinet/article?mr=0215665 MR0215665] [https://zbmath.org/?q=an%3A0153.07403 Zbl 0153.07403] </td></tr></table> ===='"`UNIQ--h-1--QINU`"'Comments==== An effective version of the result $\lim_{D \to -\infty}h(D)=\infty$ has recently been proved by B.H. Gross and D.B. Zagier [[#References|[a1]]]. ===='"`UNIQ--h-2--QINU`"'References==== <table><tr><td valign="top">[a1]</td> <td valign="top"> B.H. Gross, D.B. Zagier, "Heegner points and derivatives of $L$-series" Invent. Math. , 84 (1986) pp. 225–320 MR833192
Quadratic field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_field&oldid=24118