1911 Encyclopædia Britannica/Number/Class-Number

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

(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 &sect; 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.