Page:EB1911 - Volume 19.djvu/886

to the symbols for the classes contained in the principal genus, because two forms of that genus compound into one of the same kind. If this latter group is cyclical, that is, if all the classes of the principal genus can be represented in the form 1, 𝖠, 𝖠²,. . .𝖠^𝑣−1, the determinant 𝖣 is said to be regular; if not, the determinant is irregular. It has been proved that certain specified classes of determinants are always irregular; but no complete criterion has been found, other than working out the whole set of primitive classes, and determining the group of the principal genus, for deciding whether a given determinant is irregular or not.

If 𝖠, 𝖡 are any two classes, the total character of 𝖠𝖡 is found by compounding the characters of 𝖠 and 𝖡. In particular, the class 𝖠², which is called the duplicate of 𝖠, always belongs to the principal genus. Gauss proved, conversely, that every class in the principal genus may be expressed as the duplicate of a class. An ambiguous class satisfies 𝖠²＝1, that is, its duplicate is the principal class; and the converse of this is true. Hence if 𝖡₁, 𝖡₂,. . .𝖡_𝑖 are the base-classes for the whole composition-group, and 𝖠＝𝖡₁^𝑥 𝖡₂^𝑦 . . . 𝖡_𝒊^𝑧 (as above) 𝖠＝1, if 2𝑥＝0 or 𝑚, 2𝑦＝0 or 𝑛, &c.; hence the number of ambiguous classes is 2^𝑖. As an example, when 𝖣＝−1460, there are four ambiguous classes, represented by

(1, 0, 365), (2, 2, 183), (5, 0, 73), (10, 10, 39);

hence the composition-group must be dibasic, and in fact, if we put 𝖡₁, 𝖡₂ for the classes represented by (11, 6, 34) and (2, 2, 183), we have 𝖡₁¹⁰＝𝖡₂²＝1 and the 20 primitive classes are given by 𝖡₁^𝑥B₂^𝑦(𝑥≤10, 𝑦≤2). In this case the determinant is regular and the classes in the principal genus are 1, 𝖡₁², 𝖡₁⁴, 𝖡₁⁶, 𝖡₁⁸.

38. On account of its historical interest, we may briefly consider the form 𝑥²+𝑦², for which 𝖣＝−4. If 𝑝 is an odd prime of the form 4𝑛+1, the congruence 𝑚²≡−4(mod 4𝑝) is soluble (§ 31); let one of its roots be 𝑚, and 𝑚²+4＝4𝑙𝑝. Then (𝑝, 𝑚, 𝑙) is of determinant −4, and, since there is only one primitive class for this determinant, we must have (𝑝, 𝑚, 𝑙)~(1, 0, 1). By known rules we can actually find a substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ which converts the first form into the second; this being so, ${\begin{pmatrix}\delta ,&-\beta \\-\gamma ,&\alpha \end{pmatrix}}$ will transform the second into the first, and we shall have 𝑝＝γ²+δ², a representation of 𝑝 as the sum of two squares. This is unique, except that we may put 𝑝＝(±γ)²+(±δ)². We also have 2＝1²+1² while no prime 4𝑛+3 admits of such a representation.

The theory of composition for this determinant is expressed by the identity (𝑥²+𝑦²) (𝑥′²+𝑦′²)＝(𝑥𝑥′±𝑦𝑦′)²+(𝑥𝑦′∓𝑦𝑥′)²; and by repeated application of this, and the previous theorem, we can show that if 𝖭=2^𝑎𝑝^𝑏𝑞^𝑐. . ., where 𝑝, 𝑞,. . . are odd primes of the form 4𝑛+1, we can find solutions of 𝖭＝𝑥²+𝑦², and indeed distinct solutions. For instance 65＝1²+8²＝4²+7², and conversely two distinct representations 𝖭＝𝑥²+𝑦²＝𝑢²+𝑣² lead to the conclusion that 𝖭 is composite. This is a simple example of the application of the theory of forms to the difficult problem of deciding whether a given large number is prime or composite; an application first indicated by Gauss, though, in the present simple case, probably known to Fermat.

39. Number of classes. Class-number Relations.—It appears from Gauss’s posthumous papers that he solved the very difficult problem of finding a formula for ℎ(𝖣), the number of properly primitive classes for the determinant 𝖣. The first published solution, however, was that of P. G. L. Dirichlet; it depends on the consideration of series of the form Σ(𝑎𝑥²+𝑏𝑥𝑦+𝑐𝑦²)^−1−𝑠 where 𝑠 is a positive quantity, ultimately made very small. L. Kronecker has shown the connexion of Dirichlet’s results with the theory of elliptic functions, and obtained more comprehensive formulae by taking (𝑎, 𝑏, 𝑐) as the standard type of a quadratic form, whereas Gauss, Dirichlet, and most of their successors, took (𝑎, 2𝑏, 𝑐) as the standard, calling (𝑏²−𝑎𝑐) its determinant. As a sample of the kind of formulae that are obtained, let 𝑝 be a prime of the form 4𝑛+3; then

$h(-4p)=\Sigma \alpha -\Sigma \beta$, $\qquad h(4p)\log(t+u{\sqrt {p}})=\log \Pi \left(\tan {b\pi \over 4p}\right)$

where in the first formula Σα means the sum of all quadratic residues of 𝑝 contained in the series 1, 2, 3,. . .⁠1/2⁠(𝑝∼1) and Σβ is the sum of the remaining non-residues; while in the second formula (𝑡, 𝑢) is the least positive solution of 𝑡²−𝑝𝑢²＝1, and the product extends to all values of 𝑏 in the set 1, 3, 5,. . .(4𝑝−1) of which 𝑝 is a non-residue. The remarkable fact will be noticed that the second formula gives a solution of the Pellian equation in a trigonometrical form.

Kronecker was the first to discover, in connexion with the complex multiplication of elliptic functions, the simplest instances of a very curious group of arithmetical formulae involving sums of class-numbers and other arithmetical functions; the theory of these relations has been greatly extended by A. Hurwitz. The simplest of all these theorems may be stated as follows. Let 𝖧 (Δ) represent the number of classes for the determinant −Δ, with the convention that ⁠1/2⁠ and not 1 is to be reckoned for each class containing a reduced form of the type (𝑎, o, 𝑎) and ⁠1/3⁠ for each class containing a reduced form (𝑎, 𝑎, 𝑎); then if 𝑛 is any positive integer,

$\sum _{\kappa =0,\pm 1,...}H(4n-\kappa ^{2})=\Phi (n)+\Psi (n)\qquad \qquad (-2{\sqrt {n}}\leq \kappa \leq 2{\sqrt {n}})$

where Φ(𝑛) means the sum of the divisors of 𝑛, and Ψ(𝑛) means the excess of the sum of those divisors of 𝑛 which are greater than ${\sqrt {n}}$ over the sum of those divisors which are less than ${\sqrt {n}}$. The formula is obtained by calculating in two different ways the number of reduced values of 𝑧 which satisfy the modular equation J(𝑛𝑧)＝J(𝑧),

where J(𝒛) is the absolute invariant which, for the elliptic function 𝔭(𝑢; 𝑔₂, 𝑔₃) is 𝑔₂³÷(𝑔₂³−27𝑔₃²), and 𝑧 is the ratio of any two primitive periods taken so that the real part of 𝑖𝑧 is negative (see below, § 68). It should be added that there is a series of scattered papers by J. Liouville, which implicitly contain Kronecker’s class-number relations, obtained by a purely arithmetical process without any use of transcendents.

40. Bilinear Forms.—A bilinear form means an expression of the type Σα_𝑖𝑘𝑥_𝑖𝑦_𝑘 (𝑖＝1, 2,. . .𝑚; 𝑘＝1, 2,. . .𝑛); the most important case is when 𝑚＝𝑛, and only this will be considered here. The invariants of a form are its determinant [𝑎_𝑛𝑛] and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions 𝑥′＝Σα𝑥, 𝑦′＝𝚺𝛽𝑦. Every bilinear form is equivalent to a reduced form $\sum _{1}^{r}e_{i}x_{i}y_{i}$, and 𝑟＝𝑛, unless [𝑎_𝑛𝑛]＝0. In order that two forms may be equivalent it is necessary and sufficient that their invariants should be the same. Moreover, if 𝑎∼𝑏 and 𝑐∼𝑑, and if the invariants of the forms 𝑎+λ𝑐, 𝑏+λ𝑑 are the same for all values of λ, we shall have 𝑎+λ𝑐∼𝑏+λ𝑑, and the transformation of one form to the other may be effected by a substitution which does not involve λ. The theory of bilinear forms practically includes that of quadratic forms, if we suppose 𝑥_𝑖, 𝑦_𝑖 to be cogredient variables. Kronecker has developed the case when 𝑛＝2, and deduced various class-relations for quadratic forms in a manner resembling that of Liouville. So far as the bilinear forms are concerned, the main result is that the number of classes for the positive determinant 𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁＝Δ is 12{Φ(Δ)+Ψ(Δ)}+2ε, where ε is 1 or 0 according as Δ is or is not a square, and the symbols Φ, Ψ have the meaning previously assigned to them (§ 39).

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 H(Δ) in § 39. 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 𝑛 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 𝖯𝖰², where 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 𝑚²𝑛 with 𝑛≡1, ±2, ±3 (mod 8); when 𝑚＝1 and 𝑛≡3 (mod 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＝𝑥²+𝑦²+𝑧²+𝑡² is 8Φ(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 𝖥(𝑥, 𝑦, 𝑧)+𝑡²＝1, where 𝖥 is the contra variant 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 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²＝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.

42. Complex Numbers.—One of Gauss’s most important and far-reaching contributions to arithmetic was his introduction of complex