1911 Encyclopædia Britannica/Number/Class-Number

54. Class-Number.—The number of ideal classes in the field $\Omega (\surd m)$ may be expressed in the following forms:—

(i.) $m<0$:

$h={\frac {\tau }{2\Delta }}{\underset {n}{\Sigma }}\left({\frac {\Delta }{n}}\right)n\qquad (n=1,2,\dots ,-\Delta )$;

(ii.) $m>0$:

$h={\frac {1}{2\log \epsilon }}\log {\frac {\Pi \sin {\frac {b\pi }{\Delta }}}{\Pi \sin {\frac {a\pi }{\Delta }}}}$

In the first of these formulae $\tau$ is the number of units contained in $\Omega$; thus $\tau =6$ for $\Delta =-3$, $\tau =4$ for $\Delta =-4$, $\tau =2$ in other cases. In the second formula, $\epsilon$ is the fundamental unit, and the products are taken for all the numbers of the set $(1,2,\dots ,\Delta )$ for which $\left({\frac {\Delta }{a}}\right)=+1,\left({\frac {\Delta }{b}}\right)=-1$ respectively. In the ideal theory the only way in which these formulae have been obtained is by a modification of Dirichlet's method; to prove them without the use of transcendental analysis would be a substantial advance in the theory.

55. Suppose that any ideal in $\Omega$ is expressed in the form $[\omega _{1},\omega _{2}]$; then any element of it is expressible as $x\omega _{1}+y\omega _{2}$, where $x,y$ are rational integers, and we shall have $\mathrm {N} (x\omega _{1}+y\omega _{2})=ax^{2}+bxy+cy^{2}$, where $a,b,c$ are rational numbers contained in the ideal. If we put $x=\alpha x'+\beta y',y=\gamma x'+\delta y'$, where $\alpha ,\beta ,\gamma ,\delta$ are rational numbers such that $\alpha \delta -\beta \gamma =\pm 1$, we shall have simultaneously $(a,b,c)(x,y)^{2}=(a',b',c')(x',y')^{2}$ as in § 32 and also $(a',b',c')(x',y')^{2}=\mathrm {N} \{x'(\alpha \omega _{1}+\gamma \omega _{2})+y'(\beta \omega _{1}+\delta \omega _{2})\}=\mathrm {N} (x'\omega '_{1}+y'\omega '_{2})$, where $[\omega '_{1},\omega '_{2}]$ is the same ideal as before. Thus all equivalent forms are associated with the same ideal, and the numbers representable by forms of a particular class are precisely those which are norms of numbers belonging to the associated ideal. Hence the class-number for ideals in $\Omega$ is also the class-number for a set of quadratic forms; and it can be shown that all these forms have the same determinant $\Delta$. Conversely, every class of forms of determinant $\Delta$ can be associated with a definite class of ideals in $\Omega (\surd m)$, where $m=\Delta$ or ${\tfrac {1}{4}}\Delta$ as the case may be. Composition of form-classes exactly corresponds to the multiplication of ideals: hence the complete analogy between the two theories, so long as they are really in contact. There is a corresponding theory of forms in connexion with a field of order $n$: the forms are of the order $n$, but are only very special forms of that order, because they are algebraically resolvable into the product of linear factors.