U11, u12, u13 • . . U21, u22, U23 • unl, un2 • . . . .
with w.w or w2; and so on. Thus we may construct arrangements of aggregates corresponding to any index of the form 0(w) =awn+bw"1+ . . . +hw+l,
where n, a, b, . . . l are all positive integers.
We are thus led to the construction of a scheme of symbols
I. I,2,3 . . n . .
co, co+1, . . . w+n, . .
2w, 2w+I, . . . 2w+n, . .
II. .0,2,0+1,0+2 . . . w2+n,
~(w), 0(w)+I, . . . O(w)+n, . .
ww+I, . . . ww+n, . . .
wA(w), w95(w).+I, wO(w)+n,
The symbols 0(w) form a countable aggregate: so that we may, if we like (and in various ways), arrange the rows of block (II.) in a scheme of type w: we thus have each element a succeeded in its row by (a+ 1), and the row containing rb(w) succeeded by a definite next row. The same process may be applied to (III.), and we can form additional blocks (IV.), (V.), &c., with
first elements w4 = www' w5 = co' 4' &c. All the symbols in which w occurs are called transfinite ordinal numbers.
21. The index of a finite set is a definite integer however the set may be arranged; we may take this index as also denoting the power of the set, and call it the number of things in the set. But the index of an infinite ordinable set depends upon the way in which its elements are arranged; for instance, ind. (I, 2, 3, . . . )=w, but ind. (I, 3, 5, . . . 2, 4, 6, . . . ) = 2W. Or, to take another example, the scheme
I, 3, 5, . . . (2n—I) . .
2, 6, I0, . . . 2 (2n—I). . .
2m 2"".. 3, 2"`• 5, . . . 2"' (2n—I) .
where each row is supposed to follow the one above it, gives a permutation of (1, 2, 3, . . . ), by which its index is changed from w to w2. It has been proved that there is a permutation of the natural scale, of which the index is O(w), any assigned element of (II.); and that, if the index of any ordered aggregate is O(w), the aggregate is countable. Thus the power of all aggregates which can be associated with indices of the class (II.) is the same as that of the natural scale; this power may be denoted by a. Since a is associated with all aggregates of a
particular power, independently of the arrangement of their elements, it is analogous to the integers, I, 2, 3, &c., when used to denote powers of finite aggregates; for this reason it is called the least transfinite cardinal number.
22. There are aggregates which have a power greater than a: for instance, the arithmetical continuum of positive real numbers, the power of which is denoted by c. Another one is the aggregate of all those ordertypes which (like those in II. above) are the indices of aggregates of power a. The power of this aggregate is denoted by mi. According to Cantor's theory it is the transfinite cardinal number next superior to a, which for the sake of uniformity is also denoted by No. It has been conjectured that 01=c, but this has neither been verified nor disproved. The discussion of the alephnumbers is still in a controversial stage (November 1907) and the points in debate cannot be entered upon here.
23. Transfinite numbers, both ordinal and cardinal, may be combined by operations which are so far analogous to those of ordinary arithmetic that it is convenient to denote them by the same symbols. But the laws of operation are not entirely the same; for instance, 2W and W2 have different meanings: the first has been explained, the second is the index of the scheme (a, b, a2 bz a3b3 I . . . I a"b" I . . . ) or any similar arrangement. Again if n is any positive integer, na=a"=a. It should also be observed that according to Cantor's principles of construction every ordinal number is succeeded by a definite next one; but that there are definite ordinal numbers (e.g. w, co') which have no ordinal immediately preceding them.
24. Theory of Numbers.—The theory of numbers is that branch of mathematics which deals with the properties of the natural numbers. As Dirichlet observed long ago, the whole of the subject would be coextensive with mathematical analysis in general; but it is convenient to restrict it to certain fields where the appropriateness of the above definition is fairly obvious. Even so, the domain of the subject is becoming more and more comprehensive, as the methods of analysis become more systematic and more exact.
The first noteworthy classification of the natural numbers is into those which are prime and those which are composite. A prime number is one which is not exactly divisible by any number except itself and 1; all others are composite. The number of primes is infinite (Eucl Elem. ix. 20), and consequently, if n is an assigned number, however large, there is an infinite number (a) of primes greater than n.
If m, n are any two numbers, and m>n, we can always find a definite chain of positive integers (q,, (qz, rz), &c., such that
m=qin+ri, n=g2ri+r2, ri=g3r2Ir3, &c.
with n > r,> r2> r3 . . .; the process by which they are calculated will be called residuation. Since there is only a finite number of positive integers less than n, the process must terminate vdth two equalities of the form
rhz = ghrhI +rh, rn, = ghs1rh.
Hence we infer successively that rh is a divisor of rh_,, rhz,..
.
and finally of m and n. Also rh is the greatest common factor of m, n: because any common factor must divide r,, r2, and so on down to es; and the highest factor of rh is rh itself. It will be convenient to write rh=dv (m, n). If rt.= 1, the numbers m, n are said to be prime to each other, or coprimes.
25. The foregoing theorem of residuation is of the greatest importance; with the help of it we can prove three other fundamental propositions, namely:
(1) If m, n are any two natural numbers, we can always find two other natural numbers x, y such that
dv(m,n) =xm—yn.
(2) If m, n are prime to each other, and p is a prime factor of mn, then p must be a factor of either m or n.
(3) Every number may be uniquely expressed as a product of prime factors.
Hence if n = pagtlrY . . . is the representation of any number n as the product of powers of different primes, the divisors of n are the terms of the product
(1+p+p2+ . . . +pa) (I+q+ ... +q8) (1+r+ ... +rY) .. . their number is (a+I) (/3+I) (y+I) .; and their sum is
I I (pa+1 —I) = I I (p 1). This includes I and n among the divisors of n.
26. Totients.—By the totient of n, which is denoted, after Enter, by O(n), we mean the number of integers prime to n, and not exceeding n. If n = pa, the numbers not exceeding n and not prime to it arep, 2p, ... (pa—p), pa of which the number is pa': hence 0(pa) = pa—pai. If m, n are prime to each other, ¢(mn)=0(m)O(n); and hence for the general case, if n=p"g3rY . . . ,O(n)=Ilpa'(p—I), where the product applies to all the different prime factors of n. If
d2, &c., are the different divisors of n,
~(d,)+‘I) (d2)+ ... =n.
For example, 15=0(15)+0(5)+'(3)+'P(1)=8+4+2+I.
27. Residues and congruences.—It will now be convenient to include in the term " number " both zero and negative integers. Two numbers a, b are said to be congruent with respect to the modulus m, when (a—b) is divisible by m. This is expressed by the notation a—b (mod m), which was invented by Gauss. The fundamental theorems relating to congruences are
If ab and cd (mod m), then atcb 'd, and abcd.
If hahb(mod m) then ab (mod mid), where d=dv(h, m).
Thus the theory of congruences is very nearly, but not quite, similar to that of algebraic equations. With respect to a given modulus m the scale of relative integers may be distributed into m classes, any two elements of each class being congruent with respect to m. Among these will be (km) classes containing numbers prime to m. By taking any one number from each class we obtain a complete system of residues to the modulus m. Supposing (as we shall always do) that m is positive, the numbers o, I, 2, ... (m—I) form a system of least positive residues; according as m is odd or even, 0,=I,t2, . t2 (m—I), or 0,1,t2, . tZ(m—2),2m form a system of absolutely least residues.
28. The Theorems of Fermat and Wilson.—Let r1, r2, . . . re where t=¢(m), be a complete set of residues prime to the modulus m. Then if x is any number prime to m, the residues xr,, xrz, . . . xrc also form a complete set prime to m (§ 27). Consequently xr,•xr2 xr,r,r2 . . .r,, and dividing by rir2 . . . r,, which is prime to the modulus, we infer that
x'("')(mod m).
which is the general statement of Fermat's theorem. If m is a prime p, it becomes xfi'1 (mod p).
For a prime modulus p there will be among the set x, 2x, 3x, .. . (p—I)x just one and no more that is congruent to I: let this be xy. If yx, we must have x2— 1 = (x 1) (x+ I) —o, and hence x =1 : consequently the residues 2, 3, 4, ... (p—2) can be arranged in
(p—3) pairs (x, y) such that xy1. Multiplying them all together, we conclude that 2.3.4....(p—2)I and hence, since 1.(p— I) 1,
(p—I)!I (mod p),
which is Wilson's theorem. It may be generalized, like that of Fermat, but the result is not very interesting. If m is composite (m—I)!+1 cannot be a multiple of m: because m will have a prime factor p which is less than m, so that (m—I)!—o (mod p). Hence Wilson's theorem is invertible: but it does not supply any practical test to decide whether a given number is prime.
29. Exponents, Primitive Roots, Indices.—Let p denote an odd prime, and x any number prime to p. Among the powers x, x2, x3, . x"' there is certainly one, namely x"I, which
I (mod p) ; let xe be the lowest power of x such that xem I. Then e is said to be the exponent to which x appertains (mod p) : it is always a factor of (p—1) and can only be I when x1. The residues x for which e = p — I are said to be primitive roots of p. They always exist, their number is 4.(p—i), and they can be found by a methodical, though tedious, process of exhaustion. If g is any one of them, the complete set may be represented by g, g", gb, . . . &c. where a, b, &c., are the numbers less than (p—1) and prime to it, other than 1. Every number x which is prime to p is congruent, mod p, to g', where i is one of the numbers 1, 2, 3, . . . (p—1); this number i is called the index of x to the base g. Indices are analogous to logarithms: thus
indo(xy)=indgx+indoy. ind5(xh)=h index (mod p—1).
Consequently tables of primitive roots and indices for different primes are of great value for arithmetical purposes. Jacobi's Canon Arithmeticus gives a primitive root, and a table of numbers and indices for all primes less than I000.
For moduli of the forms 2p, p', 2p' there is an analogous theory (and also for 2 and 4) ; but for a composite modulus of other forms there are no primitive roots, and the nearest analogy is the representation of prime residues in the form as 13" x . . , where a, fi, y, .
are selected prime residues, and x, y, z, . . . are indices of restricted range. For Instance, all residues prime to 48 can be exhibited in the form 5a 7v 13', where x=o, 1, 2, 3; y=o, I; z=o, I; the total number of distinct residues being 4.2.2 =16 =0(48), as it should be.
30. Linear Congruences.—The congruence a'x—b' (mod in') has no solution unless dv(a', m') is a factor of b'. If this condition is satisfied, we may replace the given congruence by the equivalent one axetab (mod m), where a is prime to b as well as to m. By residuation (§§ 24, 25) we can find integers h, k such that ah—mk =1, and thence obtain xbh (mod m) as the complete solution of the given congruence. To the modulus m' there are m'lm incongruent solutions. For example, 12x30 (mod 21) reduces to 2x5 (mod 7) whence x6 (mod 7)6, 13, 2o•(mod 21). There is a theory of simultaneous
linear congruences in any number of variables, first developed with precision by Smith. Iii any particular case, It is best to replace as many as possible of the given congruences by an equivalent set obtained by successively eliminating the variables x, y, z, . . . in order. An important problem is to find a number which has given residues with respect to a given set of moduli. When possible, the solution is of the form x==a (mod m), where m is the least common multiple of the moduli. Supposing that p is a prime, and that we have a corresponding table of indices, the solution of axb (mod p) can be found by observing that ind x=ind bind a (mod pet).
31. Quadratic Residues. Law of Reciprocity.—To an odd prime
modulus p, the numbers 1, 4, 9, . . . (p1)=' are congruent to i(pi) residues only, because (px)2=x2. Thus for p=5, we have 1, 4, 9, 16°I, 4, 4, i respectively. There are therefore i(pI) quadratic residues and ,'1,(pi) quadratic nonresidues prime to p; and there is a corresponding division of incongruent classes of integers with respect top. The product of two residues or of two nonresidues is a residue; that of a residue and a nonresidue is a nonresidue; and taking any primitive root as base the index of any number is even or odd according as the number is a residue or a nonresidue. Gauss writes aRp, aNp to denote that a is a residue or nonresidue of p respectively.
Given a table of indices, the solution of x2=a(mod p) when possible, is found from 2ind x=ind a (mod p1), and the result may be written in the form x tr (mod p). But it is important to discuss the congruence x2ma without assuming that we have a table of indices. It is sufficient to consider the case x2mq (mod p), where q is a positive prime less than p; and the question arises whether the quadratic character of q with respect to p can be deduced from that of p with respect to q. The answer is contained in the following theorem, which is called the law of quadratic reciprocity (for real positive odd primes) : if p, q are each or one of them of the form 471+1, then p, q are each of them a residue, or each a nonresidue of the other; but if p, q are each of the form 4n+3, then according asp is a residue or nonresidue of q we have q a nonresidue or a residue of p.
Legendre introduced a symbol (y) which denotes + I or I ac
cording as mRq or mNq (q being a positive odd prime and m any number prime to q) ; with its help we may express the law of reciprocity in the form
(p,apzy ar...' (121) '( ) (b) NUMBER
(q/ () =(1)°cnn(sn,
This theorem was first stated by Legendre, who only partly proved it; the first complete proof, by induction, was published by Gauss, who also discovered five (or six) other more or less independent proofs of it. Many others have since been invented.
There are two supplementary theorems relating to I and 2 respectively, which may may be expressed in the form
( 1I1) =(I)f(n r, (p) =(I)I(~'),
and that (q) = (q) , if m =m' (mod q). As a simple application of
the law of reciprocity, let it be required to find the quadratic character of II with respect to 1907. We have l
\1907/ (1907) ( I6 / 1
because 6N I i. Hence 1 I R1907.
Legendre's symbol was extended by Jacobi in the following manner. Let Pybe any positive odd number, and let p, p', p", &c. be its (equal or unequal) prime factors, so that P=pp'p".... Then if Q is any number prime to P, we have a generalized symbol defined by
G95)  () (p) (p) ..
This symbol obeys the law that, if Q is odd and positive, (Q) (9) (I)I(PI)(QI),
with the supplementary laws
(=PI) (—I),(P—I) (Y) (—1)A(P2I). It is found convenient to add the conventions that (9P)=(?)
when Q and P are both odd ; and that the value of the symbol is o when P, Q are not coprimes.
In order that the congruence x2ma (mod m) may have a solution it is necessary and sufficient that a be a residue of each distinct prime factor of m If these conditions are all satisfied, and m=2KpAga...; where p, q, &c., are the distinct odd prime factors of in, being tin all, the number of incongruent solutions of the given congruence is 2e 2'41 or 2i+2, according as K<2, a =2, ore>2 respectively. The actual solutions are best found by a process of exhaustion. It should
be observed that (m) = i is a necessary but not a sufficient condition
for the possibility of the congruence.
32. Quadratic ,forms.It w ill be observed that the solution of the linear congruence ax=b (mod m) leads to all the representations of b in the form ax+my, where x, y are integers. Many of the earliest researches in the theory of numbers deal with particular cases of the problem: given four numbers in, a, b, c, it is required to find all the Integers x, y (if there be any) which satisfy the equation ax2+bxy+ cy2 =m. Fermat, for instance, discovered that every positive prime of the form 4n+i is uniquely expressible as the sum of two squares. There is a corresponding arithmetical theory for forms of any degree and any number of variables; only those of linear forms and binary quadratics are in any sense complete, as the difficulty of the problem increases very rapidly with the increase of the degree of the form considered or of the number of variables contained in it.
The form ax2}bxy+cy2 will be denoted by (a, b, c) (x, y)2 or more simply by (a, b, c) when there is no need of specifying the variables. If k is the greatest common factor of a, b, c, we may write (a, b, c) _ k(a', b', c') where (a', b', c') is a primitive form, that is, one for which dv (a', b', c') = i. The other form is then said to be derived from (a', b', c') and to have a divisor h. For the present we shall concern ourselves only with primitive forms. Writing D=b24ac, the invariant D is called the determinant of (a, b, c), and there is a first classification of forms into definite forms for which D is negative, and indefinite forms for which D is positive. The case D =o or a positive square is rejected, because in that case the form breaks up into the product of two linear factors. It will be observed that D=o, 1 (mod 4) according as b is even or odd; and that if k2 is any odd square factor of D there will be forms of determinant D and divisor k.
If we write x'=ax},By, y'=yx+Sy, we have identically
(a, b, c) (x', y')2 = (a', b', c') (x, y)2
where
a'=aa2+bay+072
b' = 2 a a fi +b( aS +liy) +2 cyS +603 +52
Hence also
D' = b'24a'c' = (aS 137)2 (02400) = (a5 t?y)2 D.
Supposing that a, 19, y, S are integers such that m507=n, a number different from zero, (a, b, c) is said to be transformed into (a', b', c') by
the substitution (a' s) of the nth order. If n2 =I, the two forms
are said to be equivalent, and the equivalence is said to be proper or improper according as n= i or n = — 1. In the case of equivalence, not only are x', y' integers wherever x, y are so, but conversely; hence every number representable by (a, b, c) is representable by (a', b', c') and conversely. For the present we shall deal with proper equivalence only and write f'f' to indicate that the forms f, f' are properly equivalent. Equivalent forms have the same divisor. A complete set of equivalent forms is said to form a class; classes of the samedivisorare said to form an order, and of these the most important is the principal order, which consists of the primitive classes. It is a fundamental theorem that for a given determinant the number of classes is finite; this is proved by showing that every class must contain one at least of a certain finite number of socalled reduced forms, which can be found by definite rules of calculation.
33. Method of Reduction.—This differs according as D is positive or negative, and will require some preliminary lemmas. Suppose that any complex quantity z=x+yi is represented in the usual way by a point (x, y) referred to rectangular axes. Then by plotting off all the points corresponding to (az+Q) / (yz+S), we obtain a complete set of properly equivalent points. These all lie on the same side of the axis of x, and there is precisely one of them and no more which satisfies the conditions: (i.) that it is not outside the area which is bounded by the lines 2x= (ii.) that it is not inside the circle x2+y2 = I ; (iii.) that it is not on the line 2X=1, or onn the arcs of the circle x2±y2 =1 intercepted by 2x=i and x=o. This point will be called the reduced point equivalent to z. In the positive halfplane (y>o) the aggregate of all reduced points occupies the interior and half the boundary of an area which will be called the fundamental triangle, because the a: eas equivalent to it and finite, are all triangles bounded by circular arcs,
and having angles o and the fundamental triangle may be considered as a special case when one vertex goes to infinity. The aggregate of equivalent triangles forms a kind of mosaic which fills up the whole of the positive halfplane. It will be convenient to denote the fundamental triangle (with its halfboundary, for which x< o) by v ; for a reason which will appear later, the set of equivalent triangles will be said to make up the modular dissection of the positive halfplane.
where p is any positive odd prime. It follows from the definition that
Now let f' = (a', b', c') be any definite form with a' positive and determinant — A. The root of a'z2+b'z+c' =o which is represented by a point in the positive halfplane is
—b'+iIo
m 2a' '
and this is a reduced point if either
(i.) b' 3b2, so that b < J aA, while 4a'<4ac< A+b2< A; in case (ii.) A=4ac—a2>3a2, or else a=b=c=,l *A; incase (iii.)A=4a2—b2>3b2,4a2=A+b2<3A,orelse a=b=c=~ A. With the help of these inequalities a complete set of reduced forms can be found by trial, and the number of classes determined. The latter cannot exceeds A ; it is in general much less.
With an indefinite form (a, b, c) we may associate the representative circle
a(x2+y2) +bx+c = o,
which cuts the axis of x in two real points. The form is said to be reduced if this circle cuts V ; the condition for this is a(a t }b +c) 4, the only solutions are t= =2, u=o; D=—3 gives (t2, 0), (=I, =I); D=—4 gives (t2, 0), (o, =I). On the other hand, if D> o the number of solutions is infinite, and if (t,, ul) is the solution for which t, u have their least positive values, all the other positive solutions may be found from
t,, +2„JD_ (ti+u2ls/ D)
(n=2, 3, 4...).
The substitutions by which (a, b, c) is transformed into itself are called its automorphs. In the case when D=o (mod 4) we have t=2T, u=2U, D=4N, and (T, U) any solution of
T2—NU2=1.
This is usually called the Pellian equation, though it should properly be associated with Fermat, who first perceived its importance. The
minimum solution can be found by converting sl N into a periodic continued fraction.
The form (a, b, c) may be improperly equivalent to itself ; in this case all its improper automorphs can be expressed in the form
, (K+bX)/2a)
(K—bX)/2c, —X 1
where 0—DX2=4ac. In particular, if bo (mod a) the form (a, b,c) is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.
36. Characters of a form or class. Genera.—Let (a, b, c) be any
primitive form; we have seen above (§ 32) that if a, 13, y, S are any integers
4(aa2+bay+cy2)(a$'+bib+cb2) =b'2 — (a&—Ry)'D
where b'=2aal3+b(aS+Ty)+2c'S. Now the expressions in brackets on the left hand may denote any two numbers m, n representable by the form (a, b, c); the formula shows that 4mn is a residue of D, and hence mn is a residue of every odd prime factor of D, and if p is
any such factor the symbols () and (p) will have the same value. Putting (a, b, c) =f, this common value is denoted by ()) and called a quadratic character (or simply character) of f with respect to p. Since a is representable by f (x =1, y = o) the value (p) is the same as (p) . For example, if D = 140, the scheme of characters for the
six reduced primitive forms, and therefore for the classes they represent, is
(5) (7) (I, o, 35) + +
(4, th2,9)
(5, 0, 7)
(3, t2, 12)
In certain cases there are supplementary characters of the type
( —f2
— 1 ) and (f) , and the characters (p) are discriminated according
as an odd or even power of p is contained in D; but in every case there are certain combinations of characters (in number onehalf of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.
For any determinant D we have a principal primitive class for which all the characters are +; this is represented by the principal form (I, o,—n) or (1, t, —n) according as D is of the form 4n or 4n+ I. The corresponding genus is called the principal genus. Thus, when D = 140, it appears from the table above that in the primitive order there are two genera, each containing three classes; and the nonexistent total characters are + — and — +.
37. Composition.Considering X, Y as given lineolinear functions of (x, y), (x', y') defined by the equations
X = poxx' +pixy' +p2x'y + payy'
Y = goxx'+pixy'+g2x'y+q,yy'
we may have identically, in x, y, x', y',
(A, B, C) (X, Y)2 = (a, b, c) (x, y)2 X (a', b', c') (x', y')'
and, this being so, the form (A, B, C) is said to be compounded of the two forms (a, b. c), (a', b', c'), the order of composition being indifferent. In order that two forms may admit of composition into a third, it is necessary and sufficient that their determinants be in the ratio of two squares. The most important case is that of two primitive forms 0, x of the same determinant; these can be compounded into a form denoted by4px or x4'which is also primitive and of the same determinant as do or x. If A, B, C are the classes to which
x, 4'x respectively belong, then any form of A compounded with any form of B gives rise to a form belonging to C. For this reason we write C =AB = BA, and speak of the multiplication or composition of classes. The principal class is usually denoted by 1, because when compounded with any other class A it gives this same class A.
The total number of primitive classes being finite, h, say, the series A, A', A', &c., must be recurring, and there will be a least exponent e such that A' =1. This exponent is a factor of h, so that every class satisfies Al' =1. Composition is associative as well as commutative,
that is to say, (AB)C=A(BC); hence the symbols Al, As,...Ah
for the h different classes define an Abelian group (see GROUPS) of order h, which is representable by one or more baseclasses BI, B2, ... Bi in such a way that each class A is enumerated once and only once by putting
A=13,13e...Be (x. m,y
with Inn ... p = h, and Bl^' = B,n = . = B;P =1. Moreover, the bases may be so chosen that m is a multiple of n, n of the next corresponding index, and so on. The same thing may be said with regard
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,..
the determinant D is said to he 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 Bi, B2,. . .Bi are the baseclasses for the whole compositiongroup, and A=Bi2B2v...B," (as above) A2 =1, if 2x =o or in, 2y =0 or in, &C.; hence the number of ambiguous classes is 2i. 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 compositiongroup must be dibasic, and in fact, if we put Bi, B2 for the classes represented by (11, 6, 34) and (2, 2, 183), we have Biio =B22= r and the 20 primitive classes are given by Bi'B21i(x610, y_ 2). In this case the determinant is regular and the classes in the principal genus are 1, Bit; Big, Bib, Bib.
38. On account of its historical interest, we may briefly consider the form x2+y2, for which D = 4. If pis 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=41p. Then (p, in, l) is of determinant 4, and, since there is only one primitive class for this determinant, we must have (p, m.1)— (r, o, I). By known rules we can actually find
a substitution (a' a) which converts the first form into the second; this being so, (S' , a) will transform the second into the first, and we
shall have p =y2+S2, a representation of p as the sum of two squares. This is unique, except that we may put p=(=7)'+(6)'. 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 re
peated application of this, and the previous theorem, we can show that if N=2°pbq°..., where p, q,... are odd primes of the form 411+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. Classnumber 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 E(ax2+bxy+cy2)r_, 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 412+3; then
h(—4p) =Ma—W, h(4p) log (t+uJ p) =log II (tamp)
where in the first formula Ea means the sum of all quadratic residues of p contained in the series 1, 2, 3,...1(p=1) and E# is the sum of the remaining nonresidues; while in the second formula (t, u) is the least positive solution of t2 —put =1, and the product extends to all values of b in the set 1, 3, 5,. ..(4P—1) of which p is a nonresidue. 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 classnumbers 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 (0) represent the number of classes for the determinant —A, with the convention that i and not i is to be reckoned for each class containing a reduced form of the type (a, o, a) and § for each class containing a reduced form (a, a, a) ; then if n is any positive integer,
E H(4n—K2) =~(n)+4'(n) (—2Jn~ KG 2s/ n)
, _o,+_i,•
where ,ls(n) means the sum of the divisors of n, and 'I'(n) means the excess of the sum of those divisors of in which are greater than J in
over the sum of those divisors which are less than J in. 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 in0ariant which, for the elliptic function p(u; g2, g3) is g23_ (g23—27g22), 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 classnumber relations, obtained by a purely arithmetical process without any use of transcendents.
40. Bilinear Forms.—A bilinear form means an expression of the type Eaikxiyk (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,,,,] and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions x' =Lax, y'=Efly. Every bilinear form is equivalent to a reduced
r
form Eeixiyi, and r =n, unless [a,,,,] =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 forrns a+Xc, b+Xd are the same for all values of A, we shall have a+Ac=b+Ad, and the transformation of one form to the other may be effected by a substitution which does not involve A. The theory of bilinear forms practically includes that of quadratic forms, if we suppose xi, yi to be cogredient variables. Kronecker has developed the case when n=2, and deduced various classrelations for quadratic forms in a manner resembling that of LiouviHe. So far as the bilinear forms are concerned, the main result is that the number of classes for the positive determinant aiia22—ai2a2i =0 is 124(()+NY(A)}+2e, where a is 1 or o according as i is or is not a square, and the symbols 43, 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 classnumber 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(0) in § 39. The complete discussion of a form requires the consideration of (12—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 classrelations 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 men with n 1, =2 =3 (mod 8) ; when m =1 and nm 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 o). 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 8k(n), if in is odd, while if in is even it is 24 times the sum of the odd factors of in. Explicit and finite, though more complicated, formulae have been obtained for the number of representations of in 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 =I, where 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 axe+bye+cz2 =o; this is always possible if —be, —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 farreaching contributions to arithmetic was his introduction of complex
integral functions of 23rd roots of unity, and let n be either of the roots. If we define an+b to be an integer, when a, b are natural numbers, the product of any number of such integers is uniquely expressible in the form In+m. Conversely every integer can he expressed as the product of a finite number of indecomposable integers a+bn, that is, integers which cannot be further resolved into factors of the same type. But this resolution is not necessarily unique: for instance 6=2.3=—n(n+I), where 2, 3, n, 77+I are all indecomposable and essentially distinct. To see the way in which Kummer surmounted the difficulty consider the congruence
u2+u+6=o(mod p)
where p is any prime, except 23. If 23Rp this has two distinct roots a1, u2; and we say that art+b is divisible by the ideal prime factor of p corresponding to u1, if aril +b=o (mod p). For instance, if p=2 we may put u1=o, u2=I and there will be two ideal factors of 2, say pi and p2 such that an+b=o (mod p1) if b=o (mod 2) and an+b=o (mod p2) if a+b=o (mod 2). If both these congruences are satisfied, a bias o (mod 2) and an+b is divisible by 2 in the ordinary sense. Moreover (an +b) (cn +d) = (bc +ad —ac)n+(bd 6ac) and if this product is divisible by pt, bd=o (mod 2), whence either an+b or c77+d is divisible by pi; while if the product is divisible by p2 we have be+ad+bd—7ac=o (mod 2) which is equivalent to (a+b) (c+d)mo (mod 2), so that again either an+b or cn+d is divisible by p2. Hence we may properly speak of p1 and P2 as prime divisors. Similarly the congruence u2+u+6=o (mod 3) defines two ideal prime factors of 3, and an+b is divisible by one or the other of these according as b=o (mod 3) or 2a+b=o (mod 3); we will call these prime factors p2, pa. With this notation we have (neglecting unit factors)
integers a+bi, where a, b are ordinary integers, and, as usual, an = 1. Ia this theory there are four units I, i; the numbers i"(a+bi) are said to be associated; a—bi is the conjugate of a+bi and we write N(atbi)=a2+b2, the norm of a+bi, its conjugate, and associates. The most fundamental proposition in the theory is that the process of residuation (§ 24) is applicable; namely, if m, n are any two complex integers and N(m)>N(n), we can always find integers q, r such that m=qn+r with N(r) ~iN(n), This may be proved analytically, but is obvious if we mark complex integers by points in a plane. Hence immediately follow propositions about resolutions into prime factors, greatest common measure, &c., analogous to those in the ordinary theory; it will only be necessary to indicate special pcints of difference.
We have 2=—i(t_+i)2, so that 2 is associated with a square. a real prime of the form 4n+3 is still a prime, but one of the form 4n+I breaks up into two conjugate prime factors, for example. 5 = (I 2i) (I +2i) An integer is even, semieven, or odd according as it is divisible by (I +02, (I +0 or is prime to (I +i). Among four associated odd integers there is one and only one which= t (mod 2+ 2i); this is said to be primary; the conjugate of a primary number is primary, and the product of any number of primaries is primary. The conditions that af•bi may be primary are b=o (mod 2) a+b—s.o (mod 4). Every complex integer can be uniquely expressed in the form i'" (I +i)"aabecr . . . , where o ~m <4, and a, b, c, . . . are primary primes.
With respect to a complex modulus m, all complex integers may be distributed into N(m) incongruous classes. If m=h(a+hi) where a, b are coprimes, we may take as representatives of these classes the residues x+yi where x=o, I, 2,...[(a2+b2)h—I}; y=o, I, 2,
.(h—I). Thus when b=o we may take x=o, 1, 2,...(h—I); y=o, I, 2,... (h—I), giving the h2 residues of the real number h; while if a+bi is prime, I, 2, 3,...(a2+b2—I) form a complete set of residues.
The number of residues of m that are prime to m is given by
¢(m) =N(m)II (I .
Np)
where the product extends to all prime factors of m. As an analogue
to Fermat 's theorem we have, for any integer prime to the modulus,
x4(m)= t (mod m),xN(p)1m I (mod p)
according as m is composite or prime. There are sti[N(p)—I} primitive roots of the prime p; a primitive root in the real theory for a real prime 4n+I is also a primitive root in the new theory for each prime factor of (4n+i), but. if p=4n+3 be a prime its primitive roots are necessarily complex.
43. If p, q are any two odd primes, we shall define the symbols
(q) rsand (q) by the congruences
/ p44
{N(q)r}° (q) rs pitN(q)—r}= (q) a (mod q),
it being understood that the symbols stand for absolutely least residues. It follows that (q) = I or — I according asp is a quadratic 2 residue of q or not; and that (q) a = I only if p is a biquadratic residue of q. If p, q are primary primes, we have two laws of reciprocity, expressed by the equations
(4) 2(1) 2' (4) a (p) 4=(—I)l{N(p)1}.l{. N(q)1}.
To these must be added the supplementary formulae (p) 2(I)}{N(e)1}, (I+2) =(I)i[(a+b)21[,
a+ln 2
(a+bi) a =i3(a1) (—) =i~{a+b—(1+b)2}
¢+bi
a+bi being a primary odd prime. In words, the law of biquadratic reciprocity for two primary odd primes maybe expressed by saying that the biquadratie characters of each prime with respect to the other are identical, unless p=4=3+2i (mod 4), in which case they are opposite. The raw of biquadratic reciprocity was discovered by Gauss, who does not seem, however, to have obtained a complete proof of it. The first published proof is that of Eiseustein, which is very beautiful and simple, but involves the theory of lemniscate functions. A proof on the lines indicated in Gauss's posthumous papers has been developed by Busche; this probably admits of simplification. Other demonstrations, for instance Jacobi's, depend on cyclotomy (see below).
44. Algebraic Numbers.—The first extension of Gauss's complex theory was made by E. E. Kummer, who considered complex numbers represented by rational integral functions of any roots of unity, thus including the ordinary theory and Gauss's as special cases. He was soon faced by the difficulty that, in some cases, the law that an integer can be uniquely expressed as the product of prime factors appeared to break down. To see how this happens take the equation n2+n+6=o, the roots of which are expressible as rational2=p2p2, 3=PiP , n=PIN2, I+n=P2pa.
Real primes of which 23 is a nonquadratic residue are also primes in the field (n); and the prime factors of any number an+b, as well as the degree of their multiplicity, may be found by factorizing (6a2—ab+b2), the norm of (an+b). Finally every integer divisible by pi is expressible in the form t 2m t (I +n)n where m, n are natural numbers (or zero); it is convenient to denote this fact by writing P2=[2, I+771, and calling the aggregate 2m+(I+n)n a compound modulus with the base 2, I +n. This generalized idea of a modulus is very important and farreaching; an aggregate is a modulus when, if a, p are any two of its elements, a+f and a—13 also belong to it. For arithmetical purposes those moduli are most useful which can be put into the form [al, a2,...a,.] which means the aggregate of all the quantities xial+x2a2+••.+xsa" obtained by assigning to (x1,x2,...x"), independently, the values ojtl, t2, &c. Compound moduli may be multiplied together, or raised to powers, by rules which will be plain from the following example. We have
P22=[4, 2(I+n), (I+77)2] =[4, 2+2n,—5+n] =[4, 12,—5+n] _ [4, 5 +771= [4, 3+771
hence
Pea=p22•p2=[4, 3+77]X[2, I+771=[8, 4+4n, 6+2n, 3+4n+7721
=[8, 4+4n, 6+277, 3+3711=(n—I)[77+2, 77—6, 31=(n—I)[I,2]. Hence every integer divisible by p23 is divisible by the actual integer (n—I) and conversely; so that in a certain sense we may regard p2 as a cube root. Similarly the cube of any other ideal prime is of the form (an+b)[i, n]. According to a principle which will be explained further on, all primes here considered may be arranged in three classes; one is that of the real primes, the others each contain ideal primes only. As we shall see presently all these results are intimately connected with the fact that for the determinant 23 there are three primitive classes, represented by (I, I, 6) (2, I, 3), (2, —I, 3) respectively.
45. Kummer's definition of ideal primes sufficed for his particular purpose, and completely restored the validity of the fundamental theorems about factors and divisibility. His complex integers were more general than any previously considered and suggested a definition of an algebraic integer in general, which is as follows : if a2 a2,...a" are ordinary integers (i.e. elements of N, § 7), and 0 satisfies an equation of the form
0" +(Ile+a20"2 + . . . +an18 +an =o,
0 is said to be an algebraic integer. We may suppose this equation irreducible; 0 is then said to be of the nth order. The n roots e, B",...B("1) are all different, and are said to be conjugate. If the equation began with aoO" instead of 0", 0 would still be an algebraic number; every algebraic number can be put into the form 0/m, where m is a natural number and 0 an algebraic integer.
Associated with 0 we have a field (or corpus) S2 = R(0) consisting of all rational functions of 0 with real rational coefficients; and in like manner we have the conjugate fields S2' =R(8'), &c. The aggregate of integers contained in St is denoted by o.
Every element of ft can be put into the form
w = co+c1B+ . . . +cn_10"1
where co, ci,...c,, are real and rational. If these coefficients are all integral, is is an integer; but the converse is not necessarily true. It is possible, however, to find a set of integers col, w2....w,, belonging to S2, such that every integer in S2 can be uniquely expressed in the form
=kiwi +hsws+... +haw"
856
where h1, h2 , ... h„ are elements of N which may be called the coordinates of w with respect to the base wt, 632, .. . w". Thus o is a modulus (§ 44), and we may write o=[wl, w2, . . . w"]. Having found one base, we can construct any number of equivalent bases by means of equations such as w;' =Ec;fwf, where the rational integral
coefficients cif are such that the determinant less] = =1.
If we write 1
wi, cos, . . . w"
J 0 = w i, w 2, . . . ui"
w"i
("1) ho1) (,,1)
w1 , ei2 , . . . wx
0 is a rational integer called the discriminant of the field. Its value is the same whatever base is chosen.
If a is any integer in 12, the product of a and its conjugates is a rational integer called the norm of a, and written N(a). By considering the equation satisfied by a we see that N(a) =aai where a, is an integer in P. It follows from the definition that if a, I t3 any two integers in S2, then N(ai?) =N(a)N(i8); and that for an ordinary real integer m, we have N(m) =m".
46. Ideals.—The extension of Kummer's results to algebraic numbers in general was independently made by J. W. R. Dedekind and Kronecker; their methods differ mainly in matters of notation and machinery, each having special advantages of its own for particular purposes. Dedekind's method is based upon the notion of an ideal, which is defined by the following properties:
(i.) An ideal m is an aggregate of integers in a
(u.) This aggregate is a modulus; that is to say, if µ, are any two elements of m (the same or different) —A' is contained in m. Hence also in contains a zero element, and µdµ' is an element of m.
(iii.) If ,s is any element of in, and to any element of o, then wµ is an element of m. It is this property that makes the notion of an ideal more specific than that of a modulus.
It is clear that ideals exist; for instance, o itself is an ideal. Again, all integers in ft which are divisible by a given integer a (in o) form an ideal; this is called a principal ideal, and is denoted by oa. Every ideal can be represented by a base (§§ 44, 45), so that we may write m = [µi, µ2, . . . µ"], meaning that every element of m can be uniquely expressed in the form Eh;µ;, where hi is a rational integer. In other words, every ideal has a base (and therefore, of course, an infinite number of bases).' If a, b are any two ideals, and ff we form the aggregate of all products a/3 obtained by multiplying each element of the first ideal by each element of the second, then this aggregate, together with all sums of such products, is an ideal which is called the product of a and b and written ab or ba. In particular oa =a, o'=0 ,oa . 01=oa/3. This law of multiplication is associative as well as commutative. It is clear that every element of ab is contained in a: it can be proved that, conversely, if every element of c is contained in a, there exists an ideal b such that ab = c. In particular, if a is any element of a, there is an ideal a' such that oa=aa'. A prime ideal is one which has no divisors except itself and o. It is a fundamental theorem that every ideal can be resolved into the product of a finite number of prime ideals, and that this resolution is unique. It is the decomposition of a principal ideal into the product of prime ideals that takes the place of the resolution of an integer into its prime factors in the ordinary theory. It may happen that all the ideals in S3 are principal ideals; in this case every resolution of an ideal into factors corresponds to the resolution of an integer into actual integral factors, and the introduction of ideals is unnecessary. But in every other case the introduction of ideals or some equivalent notion, is indispensable. When two ideals have been resolved into their prime factors, their greatest common measure and least common :aultiple are determined by the ordinary rules. Every ideal may be expressed (in an infinite number of ways) as the greatest common measure of two principal ideals.
47. There is a theory of congruences with respect to an ideal modulus. Thus a,Q (mod m) means that a$ is an element of in. With respect to in, all the integers in S2 may be arranged in a finite number of incongruent classes. The number of these classes is called the norm of in, and written N(tn). The norm of a prime ideal p is some power of a real prime p; if N(p) =pf, p is said to be a prime Ideal of degree f. If in, n are any two ideals, then N(inti) =N(m)N(n). If N(m) =m, then mo (mod m), and there is an ideal in' such that t)m =mm'. The norm of a principal ideal Va is equal to the absolute value of N(a) as defined in § 45.
The number of incongruent residues prime to in is
4,(tn) =N(m)II (I —N (p))'
where the product extends to all prime factors of in. If w is any element of n prime to in,
wd'(m)=i (mod m).
Associated with a prime modulus p for which N (p) = pf we have 4,(p/I) primitive roots, where 41 has the meaning given to it in the ordinary theory. Hence follow the usual results about exponents, indices, solutions of linear congruences, and so on. For any modulus m we have N(m) =100), where the sum extends to all the divisors of m.
48. Every element of v which is not contained in any other ideal is an algebraic unit. If the conjugate fields 0, 12', . . . S2("—1) consist of ri real and 2r2 imaginary fields, there is a system of units ei, e2, ... e„ where r=ri+r2—I, such that every unit in S2 is expressible in the form e =pei"eya ... ert where p is a root of unity contained in St and a, b, . . . l are natural numbers. This theorem is due to Dirich let. The norm of a unit is +1 or 1; and the determination of all the units contained in a given field is in fact the same as the solution of a Diophantine equation
F(hi, h2,...h.)tI.
For a quadratic field the equation is of the form hienhz2 = =1,
and the theory of this is complete; but except for certain special cubic corpora little has been done towards solving the important problem of assigning a definite process by which, for a given field, a system of fundamental units may be calculated. The researches of Jacobi, Hermite, and Minkowsky seem to show that a proper extension of the method of continued fractions is necessary.
49. Ideal Classes.—If m is any ideal, another ideal n can always be found such that inn is a principal ideal; for instance, one such multiplier is m—iN(m). Two ideals In, in' are said to be equivalent (mom') or to belong to the same class, if there is an ideal n such that Inn, tn'tt 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 S2 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 0.
If in, It are any two ideals belonging to the classes A, B respectively, then mn belongs to a definite class which depends only upon A, B and may be denoted by AB or BA indifferently. Thus the classsymbols form an Abelian group of order h, of which 0 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
3'(S) =EN(In) —e
(m)
where the sum extends to all ideals in contained in fl; this converges so long as the real quantity s is positive and greater than i. Then K being a certain quantity which can be calculated when a fundamental system of units is known, we shall have
Kh=LEI (s—));'(s)}.
s=
The expression for K is rather complicated, and very peculiar; it may be written in the form
7411.2 r'2
2 a R K w 'WA[
where [11A] means the absolute value of the square root of the discriminant of the field, r1, r2 have the same meaning as in § 48, w is the number of roots of unity in S2, and R is a determinant of the form (l;(s'),, of order (r1lr2—1), with elements which are, in a certain special sense, " logarithms " of the fundamental units Si, e2, . . .
er.
50. The discriminant A enjoys some very remarkable properties. Its value is always different from =I; there can be only a finite number of fields which have a given discriminant; and the rational prime factors of L1(S2) are precisely those rational primes which, in 12, are divisible by the square (or some higher power) of a prime ideal. Consequently, every rational prime not contained in 0 is resolvable, in 0, 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.
51. Normal Fields.—The special properties of a particular field fI are closely connected with its relations to the conjugate fields 12', 12", . 12("1). The most important case is when each of the conjugate fields is identical with S2: the field is then said to be
Galoisian or normal. The aggregate R (0, 0', . 0("i)) of all
rational functions of 0 and its conjugates is a normal field: hence every arithmetical field of order n is either normal, or contained in a normal field of a higher order. The roots of an equation f(0) =o which defines a normal field are associated with a group of substitutions: if this is Abelian, the field is called Abelian; if it is cyclic, the field is called cyclic. A cyclotomic field is one the elements of which are all expressible as rational functions of roots of unity; in particular the complete cyclotomic field Cm, of order 4,(m), is the aggregate of all rational functions of a primitive mth root of unity. To Kronecker is due the very remarkable theorem that all Abelian (including cyclic) fields are cyclotomic: the first published proof of this was given by Weber, and another is due to D. Hilbert.
Many important theorems concerning a normal field have been established by Hilbert. He shows that if S2 is a given normal field of order m, and p any of its prime ideals, there is a finite series of associated fields 121, 522, &c., of orders mi, m2, &c., such that mo=o
(mod. m'+i), and that if r' = m/m', = pi, a prime ideal in fl".
If Sti is the last of this series, it is called the field of inertia
(Traghtitskorper) for p : next of ter this comes another field of still lower order called the resolving field (Zerlegungskorper) for p, and in this field there is a prime of the first degree, pt+1, such that pt+t =pk, where k=m/mt. In the field of inertia pit' remains a prime, but becomes of higher degree; in 12t_I, which Is called the branchfield (Verzweigungskor per) it becomes a power of a prime, and by going on in this way from the resolving field to 12, we obtain (1+2) representations for any prime ideal of the resclving field. By means of these theorems, Hilbert finds an expression for the exact power to which a rational prime p occurs in the discriminant of 12, and in other ways the structure of 12 becomes more evident. It may be observed that whem m is prime the whole series reduces to 12 and the rational field, and we conclude that every prime ideal in 12 is of the first or snth degree: this is the case, for instance, when m=2, and is one of the reasons why quadratic fields are comparatively so simple in character.
52. Quadratic Fields.—Let m be an ordinary integer different from +i, and not divisible by any square: then if x, y assume all ordinary rational values the expressions x+ys/ m are the elements of a field which may be called 12(sl m). It should be observed that 'l m means one definite root of x2—m=o, it does not matter which: it is Convenient, however, to agree that I m is positive when m is positive, and i./ m is negative when m is negative. The principal results relating to 12 will now be stated, and will serve as illustrations of §§ 4451.
In the notation previously used
u=[I, 2(I+Al n1)] or [I, 11m]
according as m=i (mod 4) or not. In the first case L1=1n, in the second A =4m. The field 12 is normal, and every ideal prime in it is of the first degree.
Let q be any odd prime factor of m; then q=q2, where q is the prime ideal [q, 1(g11im)) when m=i (mod 4) and in other cases [q, s/ in]. An odd prime p of which m is a quadratic residue is the product of two prime ideals p, p', which may be written in the form 1p, i(a+'m)], [p,z(a—.lm)] or [p, a+Jm], [p,a— 'm], according as mi (mod 4) or not: here a is a root of x2=m (mod p), taken so as to be odd in the first of the two cases. All other rational odd primes are primes in 12. For the exceptional prime 2 there are four cases to consider: (i.) if mwi (mod 8), then 2=[2,1(1 Fs( m)]X[2,1(1—slm)]. (ii.) If m5 (mod 8), then 2 is prime: (iii.) if m—2 (mod 4), 2=[2m]2: (iv.) if m=3 (mod =4), 2=[2,1 ~m)2. Illustrations will be found in § 44 for the case in = 23.
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
n, m
integers, m not a square, w any rational prime; we write w ) = +i if, to the modulus w, n is congruent to the norm of an integer contained in 12(Jl m) ; in all other cases we put (n— ) = —i. This new symbol obeys a set of laws, among which may be especially noted (nu w) = (win) _ (w) and (nwi) =+1, whenever n, m are prime top. J J
Now let qt, q2, . . . qt be the different rational prime factors of the discriminant of 12(l/ m) ; then with any rational integer a we may associate the t symbols
(as m
i ) ' (agzm) ' . (ala)
n)
and call them the total character of a with respect to 11. This definition may be extended so as to give a total character for every ideal a in 12, as follows. First let 12 be an imaginary field (m o:
II sin  h=2 log E log II sin as
In the first of these formulae T is the number of units contained in 12; thus r=6 for A= 3, r =4 for Ate 4, T=2 in other cases. In the second formula, is the fundamental unit, and the products are
taken for all the numbers of the set (i,2,...Z.1) for which (6) =+I,
\b
(jt) i 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 12 is expressed in the form [wl, (02]; then any element of it is expressible as xwl+yw2, where x, y are rational integers, and we shall have N (xwi+yw2) =ax2+bxy+cy2, where a, b, c are rational numbers contained in the ideal. If we put x = ax'+i3y',y =yx'+iy', where a, /3, y, ,I are rational numbers such that ao—,By=ti, we shall have simultaneously (a, b, c) (x, y)2 = (a', b', c') (x', y')2 as in § 32 and also
(a', h', c') (x', y')2=N{x'(aw1+ywz)+Y'(/3w1+Iw2)] =N(x'ie'i+Y'w'z),
where [w'1, w'21 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 classnumber for ideals in 12 is also the classnumber for a set of quadratic farms; and it can be shown that all these forms have the same determinant 0. Conversely, every class of forms of determinant A can be associated with a definite class of ideals in 12(kl m), where m =0 or ',0 as the case may be. Composition of formclasses 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.
56. Complex Quadratic Forms.—Dirichlet, Smith and others, have discussed forms (a, b, c) in which the coefficients are complex integers of the form m+ni; and Hermite has considered bilinear forms axx'+bxy'+b'x'y+cyy', where x', y', b' are the conjugates of x, y, b and a, c, are real. Ultimately these theories are connected with fields of the fourth order; and of course in the same way we might consider forms (a, b, c) with integral coefficients belonging to any given field of order n: the theory would then be ultimately connected with a field of order 2n.
57. Kronecker's Method.—In practice it is found convenient to combine the method of Dedekind with that of Kronecker, the main principles of which are as follows. Let F( x, y, z,...) be a polynomial in any number of indeterminates (umbrae, as Sylvester calls them) with ordinary integral coefficients; if n is the greatest common measure of the coefficients, we have F = nE, where E is a primary or unit form. The positive integer n is called the divisor of F; and the divisor of the product of two forms is equal to the product of the divisors of the factors. Next suppose that the coefficients of F are integers in a field 12 of order n. Denoting the conjugate forms by F', F", ... F("1), the product FF'F" ... F("1) =fE, where f is a real positive integer, and E a unit form with real integral coefficients. The natural number f is called the norm of F. If F, G are any two forms (in 12) we have N(FG)=N(F)N(G). Let the coefficients of F be a1, a2, &c., those of G NI, /3z, &c., and those of FG ye, y2, &c.; and let p be any prime ideal in 12. Then if p"' is the highest power of p contained in each of the coefficients ai, and p" the highest power of p contained in each of the coefficients 0i, pm+" is the highest power of p contained by the whole set of coefficients yi. Writing dv(al, a2,...) for the highest ideal divisor of a1, a2, &c., this is called the content of F; and we have the theorem that the
product of the contents of two forms is equal to the content of the product of the forms. Every form isassociated with a definite ideal in, and we have N(F) =N(tn) if m is the content of F, and N(m) has the meaning already assigned to it. On the other hand, to a given ideal correspond an indefinite number of forms of which it is the content; for instance (§ 46, end) we can find forms ax+tly of which any given ideal is the content.
58. Now let col, W2, . wn be a basis of 0; u1, u2, ... tin a set of indeterminates; and
=wlul+w2u2+... +cunun:
i; is called the fundamental form of 12. It satisfies the equation N (xE) =o, or
F(x) =xn+Ulxni+ ... +Un=O
where Ul, Ha, . . . U„ are rational polynomials in u1, u2, ...un with rational integral coefficients. This is called the fundamental equation.
Suppose now that p is a rational prime, and that p=p"gore.. . where p, q, r, . &c., are the different ideal prime factors of p, then if F(x) is the lefthand side of the fundamental equation there is an identical congruence
F(x) ={P(x)I°{Q(x)Ib{R(x)I'... (mod p)
where P(x), Q(x), &c., are prime functions with respect to p. The meaning of this is that if we expand the expression on the righthand side of the congruence, the coefficient of every term xiui'".. unt will be congruent, mod p, to the corresponding coefficient in F(x). If f, g, h, &c., are the degrees of p, q, r, &c. (§ 47), then f, g, h, . are the dimensions in x, ui, u2,... u„ of the forms of P, Q, R, respectively. For every prime p, which is not a factor of 0, a=b=c=. . .=I and F(x) is congruent to the product of a set of different prime factors, as many in number as there are different ideal prime factors of p. In particular, if p is a prime in fl, F(x) is a prime function (mod p) and conversely.
It generally happens that rational integral values al, a2, . . . an can be assigned to u1, u2, . . . us such that U,,, the last term in the fundamental equation, then has a value which is prime top. Supposing that this condition is satisfied, let alwl+a2w2+ . . +a"w"=a; and let F1(a) be the result of putting x=a, ui=ai in P(x). Then the ideal p is completely determined as the greatest common divisor of p and Pi(a); and similarly for the other prime factors of p. There are, however, exceptional cases when the condition above stated is not satisfied.
59. Cyclotomy.—It follows from de Moivre's theorem that the arithmetical solution of the equation xi" —I =o corresponds to the division of the circumference of a circle into m equal parts. The case when m is composite is easily made to depend on that where m is a power of a prime; if m is a power of 2, the solution is effected by a chain of quadratic equations, and it only remains to consider the case when m =qs, a power of an odd prime. It will be convenient to write µ=(gm)=q'I(qi); if we also put r=e2m/m, the primitive roots of x"' = i will be h in number, and represented by r, r", 0, &c. where i, a, b, &c., form a complete set of prime residues to the modulus m. These will be the roots of an irreducible equation f(x)=o of degree is; the symbol f(x) denoting (x"'—i)i(x"`/Q—1). There are primitive roots of the congruence xµ =I (mod m) ; let g
be any one of these. Then if we put rah=es, we obtain all the roots of fix) =o in a definite cyclical order (ri, r2 ...rµ); and the change of r Into ro produces a cyclical permutation of the roots. It follows from this that every cyclic polynomial in ri, r2, ...rµ with rational coefficients is equal to a rational number. Thus if we write l+ag
+bg2+.+kgµ1=n, we have, in virtue of rh=ri rP...rµIkrµi=rn, and, if we use S to denote cyclical summation, S(ri rP...rµ6) = r"+rn9+... +rns', the sum of the nth powers of all the roots of f(x) =o, and this is a rational integer or zero. Since every cyclic polynomial is the sum of parts similar to S(r1"r2b...r, 1), the theorem is proved. Now let e, f be any two conjugate factors of i1., so that of =is, and let
ni=ri+ri+e+r++2,+. ..+ri+( f')e (i=l, 2,...e)
then the elementary symmetric functions Zen, 2nin1, &c., are cyclical functions of the roots of f(x) =o and therefore have rational values which can be calculated: consequently nr, n2, .. •ne, which are called the fnomial periods, are the roots of an equation
F (n) = ne+cirri+ ... +ee =o
with rational integral coefficients. This is irreducible, and defines a field of order e contained in the field defined by f(x) =o. Moreover, the change of r into re alters ni into ni+1, and we have the theorem that any cyclical function of ni, n2, . . . ne is rational. Now let h, k be any conjugate factors of f and put
zi=ri+ri+h +rises. +.i^+(fh)e (i=1, 2, 3,)
then Ih,3'l+e, )1+2e • • •yye )1+(hI)e will be the roots of an equation G(3") =)'—n1)"1+c2lh–2 + ... +eh =o,
the coefficients of which are expressible as rational polynomials in ni. Dividing h into two conjugate factors, we can deduce from G(5) =o another period equation, the coefficients of which are rational polynomials in ?12, )1, and so on. By choosing for e, h, &c., the successive prime factors of i, ending up with 2, we obtain a set of equations of
prime degree, each rational in the roots of the preceding equations, and the last having r1 and ri1 for its roots. Thus to take a very interesting historical case, let m= 17, so that j.i = i 6 = 24, the equations are all quadratics, and if we take 3 as the primitive roct of 17, they are
re+n—4=0, y2—ni I =o
2X2—2i'X+(nf —n+I 3) J= o, p2—Xp+I =0.
If two quantities (real or complex) a and b are represented in the usual way by points in a plane, the roots of x2+ax+b=o will be represented by two points which can be found by a Euclidean construction, that is to say, one requiring only the use of rule and compass. Hence a regular polygon of seventeen sides can be inscribed in a given circle by means of a Euclidean construction; a fact first discovered by Gauss, who also found the general law, which is that a regular polygon of m sides can be inscribed in a circle by Euclidean construction if and only if ¢(m) is a power of 2; in other words m=2'P where P is a product of different odd primes, each of which is of the form 2n+I.
Returning to the case m=qk, we shall call the chain of equations F(n) =o, &c., when each is of prime degree, a set of Galoisian auxiliaries. We can find different sets, because in forming them we can take the prime factors ofµ in any order we like; but their number is always the same, and their degrees always form the same aggregate, namely, the prime factors of µ. No other chain of auxiliaries having similar properties can be formed'containing fewer equations of a given prime degree p; a fact first stated by Gauss, to whom this theory is mainly due. Thus if m=q" we must have at least (K—I) auxiliaries of order q, and if qi =2' pP ..., we must also have a quadratics, fl equations of order p, and so on. For this reason a set of Galoisian auxiliaries may be regarded as providing the simplest solution of the equation f(x) =o.
6o. When m is an odd prime p, there is another very interesting way of solving the equation (xps)+ (xI)=o. As before let (rl, ra, . . be its roots arranged in a cycle by means of a primitive root of xz.1i (mod p); and let e be a primitive root of eP1 =1. Also let
B1 =ri+era+e2ra+...
6k=el+ekr2+e2kr3+•..+ekrr1 (k =2, 3,...p—2) so that Os is derived from B1 by changing a into J.
The cyclical permutation (el, r2, ...rP 1) applied to 8k converts it into a 50s; hence 0i0s/0s is unaltered, and may be expressed as a rational, and therefore as an integral function of e. It is found by calculation that we may put
919k in=P+I
ipk(e) _ end m+k Ind( p+ tern)
4+1 m=2
while
OlOp2=—p.
In the exponents of is(e) the indices are taken to the base g used to establish the cyclical order (el, 1.2 ... rp—1). Multiplying together the (p2) preceding equalities, the result is
f)1gi = —p# (€)'A (e) ...'kp3(e) = R(e)
where R(e) is a rational integral function of a the degree of which, in its reduced form, is less than 0(p— I ). Let p be any one definite root of xp1= R(e), and put B1= p: then since
Blk Bk = Y'1•Y2 ... l~ik1
we must take 8k=pk/'p1+l2 . . . bk_1=Rs(e)pk, where R5(e) is a rational function of e, which we may suppose put into its reduced integral form ; and finally, by addition of the equations which
define 81, 02, &c.,
(p— i)rl =p+R2(s)p2_jR3(e)pa+... +Rn2(s)a,'.
If in this formula we change p into e hp, and ri into rh+1, it still remains true.
It will be observed that this second mode of solution employs a Lagrangian resolvent B1; considered merely as a solution it is neither so direct nor so fundamental as that of Gauss. But the form of the solution is very interesting; and the auxiliary numbers ¢(e) have many curious properties, which have been investigated by Jacobi, Cauchy and Kronecker.
61. When m=qs, the discriminant of the corresponding cyclotomic field is iq'k, where X=q"I(Kq—K—I). The prime q is equal to qµ, where µ =0(m) =q' (q— i), and q is a prime ideal of the first degree.
If p is any rational prime distinct from g, and f the least exponent such that pf=i (mod. m), f will be a factor of p., and putting µ/f=e, we have p=pipe ...pe, where pi, pa . . . pi are different prime ideals each of the fth degree. There are similar theorems for the case when m is divisible by more than one rational prin.e.
Kummer has stated and proved laws of reciprocity for quadratic and higher residues in what are called regular fields, the definition of which is as follows. Let the field be R(e"1 ), where its is an odd prime; then this field is regular, and p is said to be a regular prime, when h, the number of ideal classes in the field, is not divisible by p. Kummer proved the very curious fact that p is regular if, and only if, it is not a factor of the denominators of the first z (p—3) Bernoullian
[k=i, z,...(p—3)]
2x(n) (2)! , (21 n)!(mod p); x=3 (mod 7).
This formula was obtained by Eisenstein, who proved it by investigating properties of integers in the field generated by re2107=o, which is a component of the field generated by seventh root.= of unity. The first formula of this kind was given by Gauss, and relates to the case p=4n+i =x2+y2; he conceals its connexion with complex numbers. Probably there are many others which have not yet been stated.
64. Higher Congruences. Functional Moduli.—Suppose that p is
a real prime, and that f(x), 0(x) are polynomials in x with rational integral coefficient:. The congruence f(x)=0(x) (mod p) is identical when each coefficient off is congruent, mod p, to the corresponding coefficient of 0. It will be convenient to write, under these circumstances, f_4(mod p) and to say that f, are equivalent, mod p. Every pclynomial of degree h is equivalent to another of equal or lower degree, which has none of its coefficients negative, and each of them less than p. Such a polynomial, with unity for the coefficient of the highest power of x contained in it, may be called a reduced polynomial with respect to p. There are, in all, ph reduced polynomials of degree h. A polynomial may or may not be equivalent to the product of two others of lower degree than itself ; in the latter case it is said to be prime. In every case, F being any polynomial, there is an equivalence Fcfifs .fl where c is an integer and fi, f2,...fi are prime functions; this resolution is unique. Moreover, it follows Irom Fermat's theorem that IF
(x))P_ F(xP),{F(x)}P2_F(XP2), and so on.
As in the case of equations, it may be proved that, when the modulus is prime, a congruence f (x)= o (mod p) cannot have more incongruent roots than the index of the highest power of x in f(x), and that if x=¢ is a solution, f(x)_(x)fi(x)'where fi(x) is another polynomial. The solutions of xP=x are all the residues of p; hence xPxx(x+r) (x+2)...(x+pi), where the righthand expression is the product of all the linear functions which are prime to p. A generalization of this is contained in the formula
x(xssinLi)_IIf(x) (mod p)
where the product includes every prime function f(x) of which the degree is a factor of m. By a process similar to that employed in finding the equation satisfied by primitive mth roots of unity, we can find an expression for the product of all prime functions of a given degree m, and prove that their number is (m> i)
• wi(pm—£pt"1s+`pm1th_...)
where a, b, c ... are the different prime factors of m. Moreover, if F is any given function, we can find a resolution FcFiF2... Fm(mod p)
859
where c is numerical, Fl is the product of ali prime linear functions which divide F, F2 is the product of all the prime quadratic factors, and so on.
65. By the functional congruence 4,(x)=,p(x) (mod p, f(x)) is meant that polynomials U, V can be found such that 0(x) =4 (x) +pU+ Vf(x) identically. We might alsowriteq,(x)4(x) (mod p, f(x)); but this is not sc, necessary here as in the preceding case of a simple modulus. Let m be the degree of f(x); without loss of generality we may suppose that the coefficient cf x"' is unity, and it will be further assumed that f(x) is a prime function, mod p. Whatever the dimensions of 0(x), there will be definite functions X(x), 4,, (x) such that 0(x)=f(x)x(x)+41(x) where 0i(x) is of lower dimension than f(x); moreover, we may suppose 0i(x) replaced by the equivalent reduced function 02(x) mod p. Finally then, 0=02 (mod p, f(x)) where $2 is a reduced function, mod ,, of order not greater than (mf). If we put p"' =n, there will be rn all (including zero) n residues to the compound modulus (p, f): let us denote these by R1, R2, ... R". Then (cf. § 28) if we reject the one zero residue (R", suppose) and take any functions of which the residue is not zero, the residues of
. . . tbR„1 will all be different, and we conclude that
(mod p, f). Every function therefore satisfies 4,"¢ (mod p, f) ; by putting ¢ =x we obtain the principal theorem stated in § 64.
A still more comprehensive theory of compound moduli is due to Kronecker; it will be sufficiently illustrated by a particular case. Let m be a fixed natural number; X, Y, Z, T assigned polynomials, with rational integral coefficients, in the independent variables x, y, z ; and let U be any polynomial of the same nature as X, Y, Z, T. We may write Uo (mod m, X, Y, Z, T) to express the fa,t that there are integral polynomials M, X', Y', Z', T' such that
U =mM +X'X+Y'Y+Z'Z+T'T
identically. In this notation UV means that UVo. The number of independent variables and the number of functions in the modulus are unrestricted; there may be no number m in the modulus, and there need not be more than one. This theory of Kronecker's is admirably adapted for the discussion of all algebraic problems of an arithmetical character, and is certain to attain a high degree of development.
It is worth mentioning that one of Gauss's proofs of the law of quadratic reciprocity (Gott. Nadir. 1818) involves the principle of a compound modulus.
66. Forms of Higher Degree.—Except for the case alluded to at the end of § 55, the theory of forms of the third and higher degree is still quite fragmentary. C. Jordan has proved that the class number is finite. H. Poincare has discussed the classification of ternary and quaternary cubics. With regard to the ternary cubic it is known that from any rational solution of f =o we can deduce another by a process which is equivalent to finding the tangential of a point (xi, yl, z4) on the curve, that is, the point where the tangent at (xi, yi, z1) meets the curve again. We thus obtain a series of solutions (Xi, yi, zr), (X2, Y2, z2), &C.. which may or may not be periodic. E. Lucas and J. J. Sylvester have proved that for certain cubics f =o has no rationa solutions; for instance x3+y3Az3=o has rational solutions only if A = ab(a+b)/c3, where a, b, c are rational integers. Waring asserted that every natural number can be expressed as the sum of not more than 9 cubes, and also as the sum of not more than 19 fourth powers; these propositions have been neither proved nor disproved.
67. Results derived from Elliptic and Theta Functions.—For the
sake of reference it will be convenient to give the expressions for the four Jacobian theta functions. Let is be any complex quantity such that the real part of iw is negative ; and let q =e'"i'°. Then
+0'
Boo(v) =Ege2ev,nio = +2q cos 2irr1+2q4 cos 47rv+20 cos 62rv+ . .
o"
a
=II(ig2n)(1+2g2ai cos 21rv+q4a2),
Boi(v) =12q cos 22rv+2q4 cos 47ry 2qa cos 6srv+ .. . = II (i  qu) (1 2g2er cos 22ry +gaa—2) , Bio(v) =2q} cos 2rv+ 2q1 cos 32rv+2q"5 COS STD+ .
=2q} cos srv[(I—q2')(1+2q2. cos 22rv+q4'),
012(v) =2q1 sin WV2q' sin 3rrv+2q't sin 57rv.. =2q; sin irv1(1q2a) (12q2' cos 2irv+g4s).
Instead of Boo(o), &c., we write Boo, &c. Clearly Bn=o; we have the important identities ~
0u' _ irOooO1oOo1 Boo' = Om' f elo4
where 0u' means the value of do11(v)/dv for v = o. If, now, we put
y K=B~, ~K —L, u=7rvoo2v,
000
numbers. He also succeeded in showing that in the field R(e2ni/P) the equation aP+13P+yP =o has no integral solutions whenever h is not divisible by p2. What is known as the " last " theorem of Fermat is his assertion that if m is any natural number exceeding 2, the equation x"'+ym =z'" has no rational solutions, except the obvious ones for which xyz=o. It would be sufficient to prove Fermat's theorem for all prime values of m; and whenever Kummer's theorem last quoted applies, Fermat's theorem will hold. Fermat's theorem is true for all values of m such that 2 d or a=d. Hence if v is the order of G(J), so that its expansion in q begins with a term in q v' we must have
i, (1.d) +Z (a•d) =Ed+Ea
d>Vn d d>Vna>Vn
d>Vn =21d
extending to all divisors of n which exceed 1/ n. Comparing this with the other value, we have
EH (4n —K2) = 22d +e" _'F(n) +tI'(n),
g d>Vn
as stated in § 39.
70. Each of the singular moduli which are the roots of G(J) =o corresponds to exactly one primitive class of definite quadratic forms, and conversely.
Corresponding to every given negative determinant 0 there is an irreducible equation gl) =o, where j =1728J, the coefficients of which are rational integers, and the degree of which is h(—A). The coefficient of the highest power of j is unity, so that j is an arithmetical integer, and its conjugate values belong one to each primitive class of determinant —A. By adjoining the square roots of the prime factors of A the function,p(j) may be resolved into the product of as many factors as there are genera of primitive classes, and the degree of each factor is equal to the number of classes in each genus. In particular, if It, I, I(i +i)} is the only reduced form for the determinant —A, the value of j is a real negative rational cube. At
the same time its approximate value is exp [ — tai• 1 +2~ ~] +744 =
744—e"V o so that, approximately, e"v 0 =m3+744 where m is a rational integer. For instance ery "=884736743.9997775 . . •= 96o'+744 very nearly, and for the class (1, i, iI) the exact value of j is96o3. Four and only four other similar determinants are known to exist, namely —II, 19, 67, 163, although thousands have been classified. According to Hermite the decimal part of e"V 163 begins with twelve nines; in this case Weber has shown that the exact value of j is218, 3.53.233.293
71. The function j(w) is the most fundamental ni a set of quantities called classinvariants. Let (a, b, c) be the representative of any class of definite quadratic forms, and let w be the root of axe+bx+c=o which has a positive imaginary part; then F (w) is said to be a class
invariant for (a, b, c) if F a +L) =F(w) for all real integers a 'Yw+S/
f3, y, S such that aS—fly =l. This is true for j(w) whatever w may be, and it is for this reason that j is so fundamental. But, as will be seen from the above examples, the value of j soon becomes so large that its calculation is impracticable. Moreover, there is the difficulty of constructing the modular equation fl(J, J') =o (§69), which
has only been done in the cases when n=2, 3 (the latter by Smith in Proc. Lond. Math. Soc. ix. p. 242).
For moderate values of 0 the difficulty can generally be removed by constructing algebraic functions of j. Suppose we have an irreducible equation
xm+clxmI ~ ... +cm=o,
the coefficients of which are rational functions of j(w). If we apply any modular substitution e,'=S(w), this leaves the equation unaltered, and consequently only permutates the roots among themselves: thus if xl(w) is any definite root we shall have xi(w') _ xi(w), where i may or may not be equal to 1. The group of unitary substitutions which leave all the roots unaltered is a factor of the complete modular group. If we put y =x(nw), y will satisfy an equation similar to that which defines x, with j' written for j; hence, since j, j' are connected by the equation fl(j, j') = o, there will be an equation %1,(x, y) =0 satisfied by x and y. By suitably choosing x we can in many cases find '(x, y) without knowing fl(', j') ; and then the equation ,,&(x, x) =o defines a set of singular moduli, each one of which belongs to a certain value of w and all the quantities derived from it by the substitutions which leave x(w) unaltered.
As one of the simplest examples, let n=2, x3j(w)=y3j(w')=o. Then the equation connecting x, y in its complete form is of the ninth degree in each variable; but it can be proved that it has a rational factor, namely
Y3—x2Y2+495xY+x324.33.53=o,
and if in this we put x=y=u, the result is
n4–2u3–495U2+24.33.53 =0,
the roots of which are 12, 20, 15, 15. It remains to find the values of w, to which they belong. Writing 72(w) = 1 j, it is found that we may define 72 in such a way that 72(w+I) =e 2'ri/372(w), 72(c o1) =72(w), whence it is found that
(aw+$) (Yb+Yai ~sPa
7z 7w+3 =e 2s 72(w).
We shall therefore have 72(2w) =72(w) for all values of such that aw+li
2w = a3—07 = I, 73+7a+$3—3372
o (mod 3). 7w+3
Putting (a, 0, 7, I) =(o, r, 1, o) the conditions are satisfied, and 2w =i'J 2. Now Ai) =1725, so that 72(i) =12; and since j(w) is positive for a pure imaginary, 72(is'2) =20. The remaining case is settled by putting
w aw
2 7w+3
with a, 13, y, 3 satisfying the same conditions as before. One solution is (I, 2, I, I) and hence w2+3w+4=o, so that 72 (321 7) =15.
Besides 72, other irrational invariants which have been used with effect are 73=A/ (j1723), the moduli K, K', their square and fourth roots, the functions f, fi, f2 defined rby
i=2l(KK')–I . fi=VK'.f,f>=VK.f,
and the function nn(nw)/n(w) where n(w) is defined by
+°o ll
27(w)=q (—sg3s2+s=~~(3, 3) =D'2II(I—qes).
 co I
72. Another powerful method, developed by C. F. Klein and K. E. R. Fricke, proceeds by discussing the deficiency of f,(j, j') =o considered as representing a curve. If this deficiency is zero, j and j' may be expressed as rational functions of the same parameter, and this replaces the modular equation in the most convenient manner. For instance, when n=7, we may put
(r2+13r+49) (r2+5r+I)3 , ,
rr' = 49.
The corresponding singular moduli are found by solving 4(r) _ cb(r'). For deficiency 1 we may find in a similar way two auxiliary functions x, y connected by some simple equation i (x, y) =0 not exceeding the fourth degree, and such that j, j' are each rational functions of x and y.
Hurwitz has extended this field of research almost indefinitely, not only by generalising the formulae for classnumber sums, such as that in § 69, but also by bringing the modularfunction theory into connexion with that of algebraic correspondence and Abelian integrals. A comparatively simple example may help to indicate the nature of these researches. From the formulae given at the beginning of § 67, we can deduce, by actual multiplication of the corresponding series, r
4'u800=B2ooOoiO,o= +~ (TI) Itiq 2/4X ~q+72 L+1=o, I tI, t2,
~Ex(m)q'"/4 [m=I, 5, 9, .. where
x(m) =E (—I) ICI
extended over all the representations m=t;2+4n2. In a similar way
!O'n0io =000102001= 2Z ( — I )i(m2) x(m)gm/2
O'nOoi=OooOio9oi2= (—I)¢(mI)x(m)gm/a sr
If, now, we write
71(w) =E(—I)I(mI)x(m) m
j3(w) =2Ex(mn)gm/4
we shall have
djl:dj2: dj3 =Oio:8ol:800
where Olo, 0o1, Boo, are connected by the relation (§ 67) 8104+0014—8004=0
which represents, in homogeneous coordinates, a quartic curve of deficiency 3. For this curve, or any equivalent algebraic figure, j,(w), j2(w) and j3(w) supply an independent set of Abelian integrals of the first kind. If we put x =V y =Ilk', it is found that
f y3=ija(w), J)=zj2(w), ) yx=2J1(w),
so that the integrals which the algebraic theory gives in connexion with x4+y4—I =o are directly identified with ji(w), j2(w), j3(w), provided that we put x= K(w).
Other functions occur in this theory analogous to ji(w), but such that in the qseries which are the expansions of them the coefficients and exponents depend on representations of numbers by quaternary quadratic forms.
73. In the Berliner Sitzungsberichte for the period 1883189o, L. Kronecker published a very important series of articles on elliptic functions, which contain many arithmetical results of extreme elegance; some of these Kronecker had announced without proof many years before. A few will be quoted here, without any attempt at demonstration; but in order to understand them, it will be necessary to bear in mind two definitions. The first relates to the LegendreJacobi symbol (b).. If a, b have a common factor we put
o; while if a is odd and b=20c, where c is odd, we put b
(b) = (¢) (¢) . The other definition relates to the classification of discriminants of quadratic forms. If D is any number that can be such a discriminant, we must have D=o or i (mod. 4), and in every
case we can write D =DoQ2, where Q2 is a square factor of D, and Do satisfies one of the following conditions, in which P denotes a product of different odd primes:
Do = P, with P= I (mod 4)
Do=4P, P=  1 (mod 4)
D0=8P, P==i (mod 4)
Numbers such as. Do are called fundamental discriminants; every discriminant is uniquely expressible as the product of a fundamental discriminant and a positive integral square.
Now let D1, D2 be any two discriminants, then D1D2 is also a discriminant, and we may put D1D2 = D = DoQ2, where Do is fundamental: this being done, we shall have
h=oo k=co (D DIQ2 DoQz
r E ( ) F (hk)
h=1 k=I h Is
)
_ a b,c[ (DI) + (12) II m~n (Q F(am2+bmn+cn2) where we are to take h,k=1,2,3,...+oo; min=o,'I, =2,.. =co except that, if D o, (tam+bn)T nU where (T, U) is the least positive solution of T2DU2=4. The sum E applies to a system of representative
a, b, c
primitive forms (a, b, c) for the determinant D, chosen so that a is prime to Q, and b, c are each divisible by all the prime factors of Q. A is any number prime to 2D and representable by (a, b, c); and finally r =2, 4, 6, 1 according as D <4, D = 4, D = 3 or D> o. The function F is quite arbitrary, subject only to the conditions that F(xy) =F(x)F(y), and that the sums on both sides are convergent. By putting F(x)=xIP, where P is a real positive quantity, it can be deduced from the foregoing that, if D2 is not a square, and if D1 is different from I,
rH(DiQ2)H(D2Q2) =Lt (D1) E (Q2) (am2+bmn+cn2)rP p=o a, b, c \A J m'a m
where the function H(d) is defined as follows for any discriminant d:
d—Ao H(d)=2Jdlogl—U,Id
gm/2, J2(w)=2E(—I)i(mI)x(m) t4 m qm '
h(d) meaning the number of primitive forms for the determinant d. This is a generalisation of a theorem due to Dirichlet.
Then is another formula which, in a certain sense, is the generalisation of Gauss's sums (§ 62) in cyclotomy. Let 1,(u, v) denote the function 011(u+v)=Bo1(u)e0i(v) and let D1, D2 be any two fundamental discriminants such that D1D2 is also fundamental and negative: then
re'il (Di) D2\l (2Si 2S2)
27rD:Dsl. 2(si/ (sx/'G \757' I1~2~/
I / f i) + 1 D2) 1 q'(am2+bmn+cn2)
a,b,c \ A \AJ m, n
where, on the lefthand side, we are to sum for si =1, 2, 3 .. f Dif ; and on the right we are to take a complete set of representative primitive forms (a, b, c) for the determinant D1D2, and give to m, n all positive and negative integral values such that amt+bmn+cn2 is odd. The quantity r is 2, if D1D9 < 4, =4 if D1D2 =4, T =6 if D1D2 = 3. By putting D2 = i, we obtain, after some easy transformations,
sa DI sn4sK =4J oEEqi( am2+bmn+cn2),
s= S — 2I A TB10
which holds for any fundamental discriminant A. For instance, taking w =iK'/K, and .=3, we have 8102 =2KK/2r, and Igi(m2+mn+n2)= 2KKJ 3sn 43 —; a verification is afforded by making 2K approach
Sr the value a, in which case q, a vanish, while the limit of qi/K is whence the limiting value of sn43 is that of 6qi/KJ 3, which
=6/4 / 3 =J 3/2, as it should be.
Several of Kronecker's formulae connect the solution of the Pellian equation with elliptic modular functions: one example may be given here. Let D be a positive discriminant of the form 8n+5, let (T, U) be the least solution of T2DU2=I: then, if h(D) is the number of primitive classes for the determinant D,
(T UJ D)h(D) =II (2KK')2
where the product on the right extends to a certain sixth part,of those values of 2KK' which are singular, and correspond to the field S2(JD), or in other words are connected with the class invariant j(sjD). For instance, if D =5, the equation to find (KK')2 is
F(x) =L(x) =f dx
2 5g x
(where, as in all that follows, log x is taken to the base e). This value is ultimately too large, but when x exceeds a million it is nearer the truth than the value given by Legendre's formula.
By a singularly profound and original analysis, Riemann succeeded in finding a formula, of the same type as Gauss's, but more exact for very large values of x. In its complete form it is very complicated; but, by omitting terms which ultimately vanish (for sufficiently large values of x) in comparison with those retained, the formula reduces to
F(x) =A+1(I)µmL(x:im) (m=l, 2, 3, 5, 6, 7, II, ...)
where the summation extends to all positive integral values of m which have no square factor, and µ is the number of different prime factors of m, with the convention that when m = I, ( i)' =1. The symbol A denotes a constant, namely
A I(X flJ2x(x2 d
m ) logx
and L is used in the sense given above.
P. L. Tch6bichev obtained some remarkable results on the frequency of primes by an ingenious application of Stirling's theorem. One of these is that there will certainly be (k+I) primes between a and b, provided that
a<5 2Jb~bRlog6(log b)2~4R(4k+25)6R
where R =J log 2+1d log 3+i log 5=rib log 30 =0.921292.... Fromthis may be inferred the truth of Bertrand's conjecture that there is always at least one prime between a and (2a2) if 2a>7. Tchebichev's results were generalized and made more precise by Sylvester.
The actual calculation of the number of primes in a given interval may be effected by a formula constructed and used by D. F. E Meissel. The following table gives the values of F(n) for various values of n, according to Meissel's determinations:
n F(n)
20,000 2,262
Ioo,000 9,592
500,000 41,538
1 ,000,000 78,498
Riemann's analysis mainly depends upon the properties of the function
r(s)_ Dr'' (n=1, 2, 3, . . . ),
considered as a function of the complex variable s. The above definition is only valid when the real part of s exceeds i ; but it can be generalized by writing
2 sin azP(z);(z)=2 f (x)°sldx
e~I
where the integral is taken from x 1+oo along the axis of real quantities to x=e, where s is a very small positive quantity, then round a circle of radius and centre at the origin, and finally from x=€ to x = +so along the axis of real quantities. This Junction r(z) is of great importance, and has been recently studied by von Mangoldt Landau and others.
Reference has already been made to the fact that if 1, m are coprimes the linear form lx +m includes an infinite number of primes. Now let (a, b, c) be any primitive quadratic form with a total generic character C; and let lx+m be a primitive linear form chosen so that all its values have the character C. Then it has been proved by Weber and Meyer that (a, b, c) is capable of representing an infinity of primes all of the linear form lx+m.
75. Arithmetical Functions.This term is applied to symbols such as 0(n), 4'(n), &c., which are associated with n by an intrinsic arithmetical definition. The function ,D(n) was written fn by Euler, who investigated its properties, and by proving the formula
55 +55
11(1q°) = Egi(3s'+s) deduced the result that
I —~ ( l
fn =J (n— I) +f (n—2) —f (n—5)—...
2
where on the right hand we are to take all positive values of s such that n2 (3s2+s) is not negative, and to interpret fo as n, if this term occurs. J. Liouville makes frequent use of this function in his papers, but denotes it by (n).
If the quantity x is positive and not integral, the symbol E(x) or [xi is used to denote the integer (including zero) which is obtained by omitting the fractional part of x; thus E(J 2) = I, E(O.7) =o, and so on. For some purposes it is convenient to extend the definition by putting E(x) = E(x), and agreeing that when x is a positive integer, E(x) =xi; it is then possible to find a Fourier sineseries representing xE(x) for all real values of x. The function E(x) has many curious and important properties, which have been investigated by Gauss, Hermite, Hacks, Pringsheim, Stern and others. What is perhaps the simplest proof of the law of quadratic reciprocity depends upon the fact that if p, q are two odd primes, and we put p=211+1, q=2k+I
End of Article: U11
