1911 Encyclopædia Britannica/Number/Normal Residues

53. Normal Residues. Genera.—Hilbert has introduced a very convenient definition, and a corresponding symbol, which is a generalization of Legendre’s quadratic character. Let $n,m$ be rational integers, $m$ not a square, $w$ any rational prime; we write $\left({\frac {n,m}{w}}\right)=+1$ if, to the modulus $w$, $n$ is congruent to the norm of an integer contained in $\Omega (\surd m)$; in all other cases we put $\left({\frac {n,m}{w}}\right)=-1$. This new symbol obeys a set of laws, among which may be especially noted $\left({\frac {n,w}{w}}\right)=\left({\frac {w,n}{w}}\right)=\left({\frac {n}{w}}\right)$ and $\left({\frac {n,m}{w}}\right)=+1$ whenever $n,m$ are prime to p.

Now let $q_{1},q_{2},\dots q_{t}$ be the different rational prime factors of the discriminant of $\Omega (\surd m)$; then with any rational integer $a$ we may associate the $t$ symbols

$\left({\frac {a,m}{q_{1}}}\right),\left({\frac {a,m}{q_{2}}}\right),\dots \left({\frac {a,m}{q_{t}}}\right)$

and call them the total character of $a$ with respect to $\Omega$. This definition may be extended so as to give a total character for every ideal ${\mathfrak {a}}$ in $\Omega$, as follows. First let $\Omega$ be an imaginary field $(m<0)$; we put $r=t,{\bar {n}}=\mathrm {N} ({\mathfrak {a}}$), and call

$\left({\frac {{\bar {n}},m}{q_{1}}}\right),\dots \left({\frac {{\bar {n}},m}{q_{t}}}\right)$

the total character of ${\mathfrak {a}}$. Secondly, let $\Omega$ be a real field; we first determine the $t$ separate characters of $-1$, and if they are all positive we put ${\bar {n}}=+\mathrm {N} ({\mathfrak {a}}),r=t$, and adopt the $r$ characters just written above as those of ${\mathfrak {a}}$. Suppose, however, that one of the characters of $-1$ is negative; without loss of generality we may take it to be that with reference to $q_{t}$. We then put $r=t-1,{\bar {n}}=\pm \mathrm {N} ({\mathfrak {a}})$ taken with such a sign that $\left({\frac {{\bar {n}},m}{q_{t}}}\right)=+1$, and take as the total character of ${\mathfrak {a}}$ the symbols $\left({\frac {{\bar {n}},m}{q_{t}}}\right)$ for $i=1,2,\dots (t-1)$.

With these definitions it can be proved that all ideals of the same class have the same total character, and hence there is a distribution of classes into genera, each genus containing those classes for which the total character is the same (cf. § 36).

Moreover, we have the fundamental theorem that an assigned set of $r$ units $\pm 1$ corresponds to an actually existing genus if, and only if, their product is +1, so that the number of actually existing genera is $2^{r-1}$. This is really equivalent to a theorem about quadratic forms first stated and proved by Gauss; the same may be said about the next proposition, which, in its natural order, is easily proved by the method of ideals, whereas Gauss had to employ the theory of ternary quadratics.

Every class of the principal genus is the square of a class.

An ambiguous ideal in $\Omega$ is defined as one which is unaltered by the change of $\surd m$ to $-\surd m$ (that is, it is the same as its conjugate) and not divisible by any rational integer except $\pm 1$. The only ambiguous prime ideals in $\Omega$ are those which are factors of its discriminant. Putting $\Delta ={{\mathfrak {q}}_{1}}^{2}{{\mathfrak {q}}_{2}}^{2}\dots {{\mathfrak {q}}_{t}}^{2}$, there are in $\Omega$ exactly $2^{t}$ ambiguous ideals: namely, those factors of $\Delta$, including ${\mathfrak {o}}$, which are not divisible by any square. It is a fundamental theorem, first proved by Gauss, that the number of ambiguous classes is equal to the number of genera.