1911 Encyclopædia Britannica/Number/Ideal Classes

49. Ideal Classes.—If ${\mathfrak {m}}$ is any ideal, another ideal ${\mathfrak {n}}$ can always be found such that ${\mathfrak {mn}}$ is a principal ideal; for instance, one such multiplier is ${\mathfrak {m}}^{-1}\mathrm {N} ({\mathfrak {m}})$. Two ideals ${\mathfrak {m}},{\mathfrak {m}}'$ are said to be equivalent (${\mathfrak {m}}\sim {\mathfrak {m}}'$) or to belong to the same class, if there is an ideal ${\mathfrak {n}}$ such that ${\mathfrak {mn}},{\mathfrak {m'n}}$ are both principal ideals. It can be proved that two ideals each equivalent to a third are equivalent to each other and that all ideals in $\Omega$ may be distributed into a finite number, $h$, of ideal classes. The class which contains all principal ideals is called the principal class and denoted by $\mathrm {O}$.

If ${\mathfrak {m}},{\mathfrak {n}}$ are any two ideals belonging to the classes $\mathrm {A} ,\mathrm {B}$ respectively, then ${\mathfrak {mn}}$ belongs to a definite class which depends only upon $\mathrm {A} ,\mathrm {B}$ and may be denoted by $\mathrm {AB}$ or $\mathrm {BA}$ indifferently. Thus the class-symbols form an Abelian group of order $h$, of which $\mathrm {O}$ is the unit element; and, mutatis mutandis, the theorems of § 37 about composition of classes still hold good.

The principal theorem with regard to the determination of $h$ is the following, which is Dedekind’s generalization of the corresponding one for quadratic fields, first obtained by Dirichlet. Let

$\zeta (s)={\underset {({\mathfrak {m}})}{\Sigma }}\mathrm {N} ({\mathfrak {m}})-^{s}$

where the sum extends to all ideals ${\mathfrak {m}}$ contained in $\Omega$; this converges so long as the real quantity $s$ is positive and greater than $1$. Then $\kappa$ being a certain quantity which can be calculated when a fundamental system of units is known, we shall have

$\kappa h={\underset {s=1}{Lt}}\{(s-1)\zeta (s)\}$.

The expression for $\kappa$ is rather complicated, and very peculiar; it may be written in the form

$\kappa ={\frac {2^{r_{1}+r_{2}}\pi ^{r_{2}}}{w}}.{\frac {\mathrm {R} }{|\surd \Delta |}}$

where $|\surd \Delta |$ means the absolute value of the square root of the discriminant of the field, $r_{1},r_{2}$ have the same meaning as in § 48, $w$ is the number of roots of unity in $\Omega$, and $\mathrm {R}$ is a determinant of the form $|l_{i}(\epsilon ^{j})|$, of order $(r_{1}+r_{2}-1)$, with elements which are, in a certain special sense, “logarithms” of the fundamental units $\epsilon _{1},\epsilon _{2},\dots \epsilon _{r}$.

50. The discriminant $\Delta$ enjoys some very remarkable properties. Its value is always different from $\pm 1$; there can be only a finite number of fields which have a given discriminant; and the rational prime factors of $\Delta (\Omega )$ are precisely those rational primes which, in $\Omega$, are divisible by the square (or some higher power) of a prime ideal. Consequently, every rational prime not contained in $\Delta$ is resolvable, in $\Omega$, into the product of distinct primes, each of which occurs only once. The presence of multiple prime factors in the discriminant was the principal difficulty in the way of extending Kummer’s method to all fields, and was overcome by the introduction of compound moduli—for this is the common characteristic of Dedekind’s and Kronecker’s procedure.