1911 Encyclopædia Britannica/Number/Higher Quadratic Forms

41. Higher Quadratic Forms.—The algebraic theory of quadratics is so complete that considerable advance has been made in the much more complicated arithmetical theory. Among the most important results relating to the general case of $n$ variables are the proof that the class-number is finite; the enumeration of the arithmetical invariants of a form; classification according to orders and genera, and proof that genera with specified characters exist; also the determination of all the rational transformations of a given form into itself. In connexion with a definite form there is the important conception of its weight; this is defined as the reciprocal of the number of its proper automorphs. Equivalent forms are of the same weight; this is defined to be the weight of their class. The weight of a genus or order is the sum of the weights of the classes contained in it; and expressions for the weight of a given genus have actually been obtained. For binary forms the sum of the weights of all the genera coincides with the expression denoted by $\mathrm {H} (\Delta )$ in § 30. The complete discussion of a form requires the consideration of $(n-2)$ associated quadratics; one of these is the contravariant of the given form, each of the others contains more than variables. For certain quaternary and senary classes there are formulae analogous to the class-relations for binary forms referred to in § 39 (see Smith, Proc. R.S. xvi., or Collected Papers, i. 510).

Among the most interesting special applications of the theory are certain propositions relating to the representation of numbers as the sum of squares. In order that a number may be expressible as the sum of two squares it is necessary and sufficient for it to be of the form $\mathrm {P} \mathrm {Q} ^{2}$, where $\mathrm {P}$ has no square factor and no prime factor of the form $4n+3$. A number is expressible as the sum of three squares if, and only if, it is of the form $m^{2}n$ with $n\equiv 1,\pm 2,\pm 3{\pmod {8}}$; when $m=1$ and $n\equiv 3{\pmod {8}}$, all the squares are odd, and hence follows Fermat's theorem that every number can be expressed as the sum of three triangular numbers (one or two of which may be 0). Another famous theorem of Fermat's is that every number can be expressed as the sum of four squares; this was first proved by Jacobi, who also proved that the number of solutions of $n=x^{2}+y^{2}+z^{2}+t^{2}$ is $8\Phi (n)$, if $n$ is odd, while if $n$ is even it is $24$ times the sum of the odd factors of $n$. Explicit and finite, though more complicated, formulae have been obtained for the number of representations of $n$ as the sum of five, six, seven and eight squares respectively. As an example of the outstanding difficulties of this part of the subject may be mentioned the problem of finding all the integral (not merely rational) automorphs of a given form $f$. When $f$ is ternary, C. Hermite has shown that the solution depends on finding all the integral solutions of $\mathrm {F} (x,y,z)+t^{2}=1$, where $\mathrm {F}$ is the contravariant of $f$.

Thanks to the researches of Gauss, Eisenstein, Smith, Hermite and others, the theory of ternary quadratics is much less incomplete than that of quadratics with four or more variables. Thus methods of reduction have been found both for definite and for indefinite forms; so that it would be possible to draw up a table of representative forms, if the result were worth the labour. One specially important theorem is the solution of $ax^{2}+by^{2}+cz^{2}=0$; this is always possible if $-bc,-ca,-ab$ are quadratic residues of $a,b,c$ respectively, and a formula can then be obtained which furnishes all the solutions.