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A quadratic form in two variables, i.e. having the form
 
A quadratic form in two variables, i.e. having the form
  
<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/b/b016/b016370/b0163701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\begin{equation}\label{eq1} \tag{* }
 +
= f (x, y)  = \
 +
ax  ^ {2} + bxy + cy  ^ {2} .
 +
\end{equation}
 +
 
 +
If  $  a, b $
 +
and  $  c $
 +
are integers, the binary quadratic form is said to be integral. The expression  $  d = ac - {b  ^ {2} } /4 $
 +
is called the discriminant or determinant of the binary quadratic form. The expression  $  b  ^ {2} - 4ac $
 +
is also sometimes referred to as the discriminant. The arithmetic theory of binary quadratic forms originated with [[Fermat, Pierre de|P. de Fermat]], who proved that any prime number of the form  $  4k + 1 $
 +
can be represented as the sum of two squares of integers. The theory of quadratic forms was completed by J.L. Lagrange and by C.F. Gauss. This theory is a special case of the theory of quadratic forms in  $  n $
 +
variables; the arithmetic theory of binary quadratic forms is equivalent to the theory of ideals in quadratic fields, and is one of the origins of algebraic number theory (cf. [[Quadratic form]]; [[Quadratic field]]).
 +
 
 +
The number of genera of binary quadratic forms with discriminant  $  d $
 +
equals  $  2 ^ {s-1 } $,
 +
where  $  s $
 +
is the number of different prime divisors of  $  d $,
 +
except for  $  d \equiv 1 $(
 +
$  \mathop{\rm mod}  4 $),
 +
$  d \equiv 0 $(
 +
$  \mathop{\rm mod}  8 $),
 +
when  $  s $
 +
is increased by one; if  $  -d $
 +
is a square, the number of different binary quadratic forms is doubled. The number  $  r(d, m) $
 +
of essentially different primitive representations of a number  $  m $
 +
by a complete system of binary quadratic forms with discriminant  $  d $
 +
is equal to the number of solutions of the equation
 +
 
 +
$$
 +
x  ^ {2}  \equiv  -d  (  \mathop{\rm mod}  m).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163703.png" /> are integers, the binary quadratic form is said to be integral. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163704.png" /> is called the discriminant or determinant of the binary quadratic form. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163705.png" /> is also sometimes referred to as the discriminant. The arithmetic theory of binary quadratic forms originated with P. Fermat, who proved that any prime number of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163706.png" /> can be represented as the sum of two squares of integers. The theory of quadratic forms was completed by J.L. Lagrange and by C.F. Gauss. This theory is a special case of the theory of quadratic forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163707.png" /> variables; the arithmetic theory of binary quadratic forms is equivalent to the theory of ideals in quadratic fields, and is one of the origins of algebraic number theory (cf. [[Quadratic form|Quadratic form]]; [[Quadratic field|Quadratic field]]).
+
As in the general case, there exists an algorithm which reduces the problem of solving a given second-degree Diophantine equation in two unknowns (in particular, an equation  $  f(x, y) = m $)
 +
to the problem on the arithmetic equivalence of two binary quadratic forms.
  
The number of genera of binary quadratic forms with discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163708.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b0163709.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637010.png" /> is the number of different prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637011.png" />, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637013.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637015.png" />), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637016.png" /> is increased by one; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637017.png" /> is a square, the number of different binary quadratic forms is doubled. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637018.png" /> of essentially different primitive representations of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637019.png" /> by a complete system of binary quadratic forms with discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637020.png" /> is equal to the number of solutions of the equation
+
All integral automorphisms of a primitive form  $  f $
 +
with  $  a \neq 0 $
 +
can be represented in the form
  
<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/b/b016/b016370/b01637021.png" /></td> </tr></table>
+
$$
 +
\left \|
  
As in the general case, there exists an algorithm which reduces the problem of solving a given second-degree Diophantine equation in two unknowns (in particular, an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637022.png" />) to the problem on the arithmetic equivalence of two binary quadratic forms.
+
\begin{array}{cc}
 +
t - bu/2  &-cu  \\
 +
au  &t + bu/2  \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
  
All integral automorphisms of a primitive form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637024.png" /> can be represented in the form
+
where  $  t  ^ {2} + du  ^ {2} = 1 $,
 +
and  $  2t $
 +
and  $  u $
 +
are integers (cf. [[Pell equation|Pell equation]]). Therefore, the problem on the equivalence of two forms is solved by the reduction theory of binary quadratic forms. The reduction theory of positive-definite binary quadratic forms is a special case of the reduction theory of positive-definite quadratic forms according to H. Minkowski. The reduction theory of integral indefinite binary quadratic forms can be reduced to the reduction theory of quadratic irrationalities ([[#References|[2]]], [[#References|[3]]]).
  
<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/b/b016/b016370/b01637025.png" /></td> </tr></table>
+
An important role in the theory of numbers is played by the arithmetic function  $  h(d) $—
 +
the number of classes of primitive integral binary quadratic forms with determinant  $  d $.
 +
It is known that  $  h(d) < + \infty $.
 +
Some idea of the rate of increase of the function  $  h(d) $
 +
can be obtained from Siegel's theorem: Let  $  d > 0 $,
 +
then for any  $  \epsilon > 0 $
 +
there exist constants  $  c _  \epsilon  $
 +
and  $  c _  \epsilon  ^  \prime  > 0 $
 +
which satisfy the condition
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637028.png" /> are integers (cf. [[Pell equation|Pell equation]]). Therefore, the problem on the equivalence of two forms is solved by the reduction theory of binary quadratic forms. The reduction theory of positive-definite binary quadratic forms is a special case of the reduction theory of positive-definite quadratic forms according to H. Minkowski. The reduction theory of integral indefinite binary quadratic forms can be reduced to the reduction theory of quadratic irrationalities ([[#References|[2]]], [[#References|[3]]]).
+
$$
 +
c _  \epsilon  ^  \prime  d ^ {1/2 - \epsilon }  < \
 +
h (d)  < c _  \epsilon  d ^ {1/2 + \epsilon }
 +
$$
  
An important role in the theory of numbers is played by the arithmetic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637029.png" /> — the number of classes of primitive integral binary quadratic forms with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637030.png" />. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637031.png" />. Some idea of the rate of increase of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637032.png" /> can be obtained from Siegel's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637033.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637034.png" /> there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637036.png" /> which satisfy the condition
+
(a similar formula is also valid for $  d < 0 $).
  
<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/b/b016/b016370/b01637037.png" /></td> </tr></table>
+
Let  $  \Delta $
 +
be an integer,  $  \Delta \equiv 1 $
 +
or  $  0 $(
 +
$  \mathop{\rm mod}  4 $),
 +
suppose that if  $  s  ^ {2} \mid  \Delta $
 +
then  $  s = 1 $
 +
or  $  s = 2 $,
 +
and let  $  F = \mathbf Q ( \sqrt \Delta ) $
 +
be the quadratic field which is obtained by adjoining  $  \sqrt \Delta $
 +
to the field of rational numbers. A correspondence has been established between the integral ideals  $  [ \alpha _ {1} , \alpha _ {2} ] $
 +
of the field  $  F $
 +
and the integral quadratic forms
  
(a similar formula is also valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637038.png" />).
+
$$
 +
f (x, y)  = \
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637039.png" /> be an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637041.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637042.png" />), suppose that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637043.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637044.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637045.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637046.png" /> be the quadratic field which is obtained by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637047.png" /> to the field of rational numbers. A correspondence has been established between the integral ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637048.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637049.png" /> and the integral quadratic forms
+
\frac{N ( \alpha _ {1} x + \alpha _ {2} y) }{N [ \alpha _ {1} , \alpha _ {2} ] }
  
<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/b/b016/b016370/b01637050.png" /></td> </tr></table>
+
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637051.png" /> as determinant. This results in a one-to-one correspondence (up to a transition to conjugate classes of ideals) between the ideal classes of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637052.png" /> and the classes of binary quadratic forms. In this correspondence, multiplication of ideal classes defines a composition of classes of binary quadratic forms.
+
with $  - \Delta /4 $
 +
as determinant. This results in a one-to-one correspondence (up to a transition to conjugate classes of ideals) between the ideal classes of the field $  F $
 +
and the classes of binary quadratic forms. In this correspondence, multiplication of ideal classes defines a composition of classes of binary quadratic forms.
  
As in the case of forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637053.png" /> variables, the theory of binary quadratic forms can be generalized to include forms (*) with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637055.png" /> in a given algebraic number field.
+
As in the case of forms in $  n $
 +
variables, the theory of binary quadratic forms can be generalized to include forms \eqref{eq1} with coefficients $  a, b $
 +
and $  c $
 +
in a given algebraic number field.
  
There are various variants in the definitions of an integral form, the discriminant of the form, equivalence of forms, and the classes and the genera of forms. The definition of integral forms given above is due to L. Kronecker. Gauss [[#References|[1]]] stipulated that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637056.png" /> be even. In determining the equivalence (and the class of forms), only the substitutions with discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637057.png" /> may be considered; in other cases discriminants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016370/b01637058.png" /> are considered. The definition of a genus given in [[#References|[6]]] is wider than that given by Gauss.
+
There are various variants in the definitions of an integral form, the discriminant of the form, equivalence of forms, and the classes and the genera of forms. The definition of integral forms given above is due to L. Kronecker. Gauss [[#References|[1]]] stipulated that b $
 +
be even. In determining the equivalence (and the class of forms), only the substitutions with discriminant $  + 1 $
 +
may be considered; in other cases discriminants $  \pm  1 $
 +
are considered. The definition of a genus given in [[#References|[6]]] is wider than that given by Gauss.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin) {{MR|0197380}} {{ZBL|0136.32301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0265267}} {{ZBL|0204.37101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.W. Jones, "The arithmetic theory of quadratic forms" , Math. Assoc. Amer. (1950) {{MR|0037321}} {{ZBL|0041.17505}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian) {{MR|201368}} {{ZBL|0142.01403}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Landau, "Vorlesungen über Zahlentheorie" , '''3''' , Hirzel (1927) {{MR|0250844}} {{ZBL|53.0123.17}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) {{MR|}} {{ZBL|0259.10018}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin) {{MR|0197380}} {{ZBL|0136.32301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0265267}} {{ZBL|0204.37101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.W. Jones, "The arithmetic theory of quadratic forms" , Math. Assoc. Amer. (1950) {{MR|0037321}} {{ZBL|0041.17505}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian) {{MR|201368}} {{ZBL|0142.01403}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Landau, "Vorlesungen über Zahlentheorie" , '''3''' , Hirzel (1927) {{MR|0250844}} {{ZBL|53.0123.17}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) {{MR|}} {{ZBL|0259.10018}} </TD></TR></table>

Latest revision as of 15:17, 31 March 2024


A quadratic form in two variables, i.e. having the form

\begin{equation}\label{eq1} \tag{* } f = f (x, y) = \ ax ^ {2} + bxy + cy ^ {2} . \end{equation}

If $ a, b $ and $ c $ are integers, the binary quadratic form is said to be integral. The expression $ d = ac - {b ^ {2} } /4 $ is called the discriminant or determinant of the binary quadratic form. The expression $ b ^ {2} - 4ac $ is also sometimes referred to as the discriminant. The arithmetic theory of binary quadratic forms originated with P. de Fermat, who proved that any prime number of the form $ 4k + 1 $ can be represented as the sum of two squares of integers. The theory of quadratic forms was completed by J.L. Lagrange and by C.F. Gauss. This theory is a special case of the theory of quadratic forms in $ n $ variables; the arithmetic theory of binary quadratic forms is equivalent to the theory of ideals in quadratic fields, and is one of the origins of algebraic number theory (cf. Quadratic form; Quadratic field).

The number of genera of binary quadratic forms with discriminant $ d $ equals $ 2 ^ {s-1 } $, where $ s $ is the number of different prime divisors of $ d $, except for $ d \equiv 1 $( $ \mathop{\rm mod} 4 $), $ d \equiv 0 $( $ \mathop{\rm mod} 8 $), when $ s $ is increased by one; if $ -d $ is a square, the number of different binary quadratic forms is doubled. The number $ r(d, m) $ of essentially different primitive representations of a number $ m $ by a complete system of binary quadratic forms with discriminant $ d $ is equal to the number of solutions of the equation

$$ x ^ {2} \equiv -d ( \mathop{\rm mod} m). $$

As in the general case, there exists an algorithm which reduces the problem of solving a given second-degree Diophantine equation in two unknowns (in particular, an equation $ f(x, y) = m $) to the problem on the arithmetic equivalence of two binary quadratic forms.

All integral automorphisms of a primitive form $ f $ with $ a \neq 0 $ can be represented in the form

$$ \left \| \begin{array}{cc} t - bu/2 &-cu \\ au &t + bu/2 \\ \end{array} \ \right \| , $$

where $ t ^ {2} + du ^ {2} = 1 $, and $ 2t $ and $ u $ are integers (cf. Pell equation). Therefore, the problem on the equivalence of two forms is solved by the reduction theory of binary quadratic forms. The reduction theory of positive-definite binary quadratic forms is a special case of the reduction theory of positive-definite quadratic forms according to H. Minkowski. The reduction theory of integral indefinite binary quadratic forms can be reduced to the reduction theory of quadratic irrationalities ([2], [3]).

An important role in the theory of numbers is played by the arithmetic function $ h(d) $— the number of classes of primitive integral binary quadratic forms with determinant $ d $. It is known that $ h(d) < + \infty $. Some idea of the rate of increase of the function $ h(d) $ can be obtained from Siegel's theorem: Let $ d > 0 $, then for any $ \epsilon > 0 $ there exist constants $ c _ \epsilon $ and $ c _ \epsilon ^ \prime > 0 $ which satisfy the condition

$$ c _ \epsilon ^ \prime d ^ {1/2 - \epsilon } < \ h (d) < c _ \epsilon d ^ {1/2 + \epsilon } $$

(a similar formula is also valid for $ d < 0 $).

Let $ \Delta $ be an integer, $ \Delta \equiv 1 $ or $ 0 $( $ \mathop{\rm mod} 4 $), suppose that if $ s ^ {2} \mid \Delta $ then $ s = 1 $ or $ s = 2 $, and let $ F = \mathbf Q ( \sqrt \Delta ) $ be the quadratic field which is obtained by adjoining $ \sqrt \Delta $ to the field of rational numbers. A correspondence has been established between the integral ideals $ [ \alpha _ {1} , \alpha _ {2} ] $ of the field $ F $ and the integral quadratic forms

$$ f (x, y) = \ \frac{N ( \alpha _ {1} x + \alpha _ {2} y) }{N [ \alpha _ {1} , \alpha _ {2} ] } $$

with $ - \Delta /4 $ as determinant. This results in a one-to-one correspondence (up to a transition to conjugate classes of ideals) between the ideal classes of the field $ F $ and the classes of binary quadratic forms. In this correspondence, multiplication of ideal classes defines a composition of classes of binary quadratic forms.

As in the case of forms in $ n $ variables, the theory of binary quadratic forms can be generalized to include forms \eqref{eq1} with coefficients $ a, b $ and $ c $ in a given algebraic number field.

There are various variants in the definitions of an integral form, the discriminant of the form, equivalence of forms, and the classes and the genera of forms. The definition of integral forms given above is due to L. Kronecker. Gauss [1] stipulated that $ b $ be even. In determining the equivalence (and the class of forms), only the substitutions with discriminant $ + 1 $ may be considered; in other cases discriminants $ \pm 1 $ are considered. The definition of a genus given in [6] is wider than that given by Gauss.

References

[1] C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin) MR0197380 Zbl 0136.32301
[2] B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) MR0265267 Zbl 0204.37101
[3] B.W. Jones, "The arithmetic theory of quadratic forms" , Math. Assoc. Amer. (1950) MR0037321 Zbl 0041.17505
[4] A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian) MR201368 Zbl 0142.01403
[5] E. Landau, "Vorlesungen über Zahlentheorie" , 3 , Hirzel (1927) MR0250844 Zbl 53.0123.17
[6] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[7] O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018
How to Cite This Entry:
Binary quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_quadratic_form&oldid=24044
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article