U11 , u12, u13 • . . U21, u22, U23 • unl, un2 • . . . . with w.w or See also:

• W2(P — R1)

. . 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 See also:

• BLOCK
• BLOCK (from the Fr. bloc, and possibly connected with an Old Ger. Block, obstruction, cf. " baulk ")
• BLOCK, MAURICE (1816–19or)

. . ) = 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 See also:

• SCALE
• SCALE (1) A

It has been conjectured that 01=c, but this has neither been verified nor disproved. The discussion of the aleph-numbers is still in a controversial See also:

• STAGE (Fr. 6/age; from Lat. stare, to stand)

Theory of Numbers.—The theory of numbers is that See also:

• BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

. and finally of m and n. Also rh is the greatest See also:

• COMMON

+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—pa-i. If m, n are prime to each other, ¢(mn)=0(m)O(n); and hence for the general See also:

• CASE
• CASE, JOHN (d. 1600)

=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 See also:

• TERM

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 See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

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 y-x, 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 xy-1. 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 See also:

• SUPPLY (through Fr. from Lat. supplere, to fill up)

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 See also:

• BASE

. , 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 See also:

• TOTAL

. . 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 ax-b (mod p) can be found by observing that ind x=ind b-ind a (mod pet). 31. Quadratic Residues. See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

Gauss writes aRp, aNp to denote that a is a residue or non-residue 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 p-1), 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 See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

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(P-I)(Q-I), with the supplementary laws (=PI) (—I),(P—I) (Y) (—1)A(P2-I). 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 co-primes. 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 See also:

• TIN (Lat.- stannum, whence the chemical symbol " Sn "; atomic weight =117.6, 0=16)

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 See also:

• PRESENT

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 See also:

• PRINCIPAL

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 half-plane is —b'+iIo m 2a' ' and this is a reduced point if either (i.) b' <a' <c' (ii.) b' =a', (iii.) a' =c', Cases (ii.) and (iii.) only occur when the representative point is on the boundary of V. A form whose representative point is reduced is said to be a reduced form, It follows from the geometrical theory that every form is equivalent to a reduced form, and that there are as many distinct classes of positive forms of determinant —A as there are reduced forms. The total number of reduced forms is limited, because in case (i.) we have A = 4ac —b2> 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 See also:

• ASSOCIATE (Lat. associates, from ad, to, and sociare to join)

34. Probtem of Representation.—It is required to find out whether a given number m' can be represented by the given form (a', b', c'). One condition is clearly that the divisor of the form must be a factor of m'. Suppose this is the case; and let m, (a, b, c) be the quotients of m' and (a', b', c') be the divisor in question. Then we have now to discover whether m can be represented by the primitive form (a, b, c). First of all we will consider proper representations m = (a, b, c) (a, y)2 where a, y are co-primes. Determine integers $, S such that aS -137 =1, and apply to (a, b, c) the substitution (y' S) ; the new form will be (m, n, l), where n2—4ml=D=b2—4ac. Consequently n' =D (mod 4m), and D must be a quadratic residue of m. Unless this condition is satisfied, there is no proper representation of m by any form of determinant D. Suppose, however, that n'=D (mod 4m) is soluble and that n,, n2, &c. are its roots. Taking any one of these, say we can find out whether (m, n;, li) and (a, b, c) are equivalent; if they are, there is a substitution 7a1S) which converts the latter into the former, and then m = See also:

• ACT (Lat. actus, actum)

35. Automorphs. The Pellian Equation.—A primitive form (a, b, c) is, by definition, equivalent to itself; but it may be so in more ways than one In order that (a, b, c) may be transformed into itself by the substitution (y' , it is necessary and sufficient that (a, /3\ _ (s(t+bu), —cu ) y, S au, 1(t—bu) where (t, u) is an integral solution of 12—Due=4. If D is negative and —D>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 See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

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 See also:

• LEFT

37. See also:

• COMPOSITION
• COMPOSITION (Lat. compositio, from componere, to put together)

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 See also:

• COMPOUND
• COMPOUND (from Lat. componere, to combine or put together)

In this case the determinant is regular and the classes in the principal genus are 1, See also:

• BIT (from the verb " to bite," either in the sense of a piece bitten off, or an act of biting, or a thing that bites or is bitten)

Number of classes. Class-number Relations.—It appears from Gauss's See also:

• POSTHUMOUS

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 See also:

• CONVENTION (Lat. conventio, an assembly or agreement, from convenire, to come together)
• CONVENTION, THE NATIONAL

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 class-relations 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 class-number is finite; the enumeration of the arithmetical invariants of a form; classification according to orders and genera, and proof that genera with specified characters exist; also the de-termination of all the rational transformations of a given form into itself. In connexion with a definite form there is the important conception of its See also:

• WEIGHT

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 See also:

• QUATERNARY

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 See also:

• WORTH
• WORTH, CHARLES FREDERICK (1825-1895)

If both these congruences are satisfied, a--- See also:

• BIAS
• BIAS (from the Fr. biais, of unknown origin; the derivation from Lat. bifax, two-faced, is wrong)

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 co-primes, 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 See also:

• BIQUADRATIC (from the Lat. bi-, bis, twice, and quadratus, squared)

Algebraic Numbers.—The first See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

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 + . . . +an-18 +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 See also:

• COL
• COL (Fr. for " neck," Lat. collum)

. . w" J 0 = w i, w 2, . . . ui" w"i ("-1) ho-1) (,,-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 µd-µ' 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 See also:

• PROPERTY

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 See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

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 m-o (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 hie--nhz2 = =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 class-symbols 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 See also:

• GOOD, JOHN MASON (1764-1827)

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 See also:

• PROCEDURE (Fr. procedure, from Lat. procedere, to go for-ward)

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 See also:

• WEBER, CARL MARIA FRIEDRICH ERNEST VON (1786–1826)
• WEBER, WILHELM EDUARD (1804-1891)

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(g-1-1im)) 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 m-i (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 m-5 (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) ' . (See also:

• ALA

Moreover, we have the fundamental theorem that an assigned set of r units t I corresponds to an actually existing genus if, and only if, their product is +1, so that the number of actually existing genera is 2'`I. This is really equivalent to a theorem about quadratic forms first stated and proved by Gauss; the same may be said about thenext proposition, which, in its natural order, is easily proved by the method of ideals, whereas Gauss had to employ the theory of ternary quadratics. Every class of the principal genus is the square of a class. An ambiguous ideal in 12 is defined as one which is unaltered by the See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

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 class-number for ideals in 12 is also the class-number 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 form-classes exactly corresponds to the multiplication of ideals: hence the complete analogy between the two theories, so long as they are really in See also:

• CON

Let F( x, y, z,...) be a polynomial in any number of indeterminates (umbrae, as See also:

• SYLVESTER, JAMES
• SYLVESTER, JOSHUA (1563–1618)

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 (x-E) =o, or F(x) =xn+Ulxn-i+ ... +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"See also:

• GORE
• GORE, CATHERINE GRACE FRANCES (1799-1861)
• GORE, CHARLES (1853– )

(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 right-hand 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(q-i); 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 See also:

• FIX, THEODORE (180o-1846)

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 f-nomial 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^+(f-h)e (i=1, 2, 3,) then Ih,3'l+e, )1+2e • • •yye )1+(h-I)e will be the roots of an equation G(3") =)'—n1)"-1+c2l-h–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 See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

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 ri-1 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 See also:

• RULE

. be its roots arranged in a See also:

• CYCLE
• CYCLE (Gr. KUK)^os, a circle)

But the form of the solution is very interesting; and the See also:

• AUXILIARY (from Lat. auxilium, help)

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 F--cfifs .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 xP-x-x(x+r) (x+2)...(x+p-i), where the right-hand 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 F-cFiF2... Fm(mod p) 859 where c is numerical, Fl is the product of See also:

• ALI

Finally then, 0=02 (mod p, f(x)) where $2 is a reduced function, mod ,, of order not greater than (m-f). 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 U-o (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 U-V means that U-V-o. 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. See also:

• NADIR (Arabic nadir, " opposite to," used elliptically for nadir-es-semt, "opposite to the zenith ")

E. See also:

• LUCAS, CHARLES (1713–1771)
• LUCAS, JOHN SEYMOUR (1849– )
• LUCAS, SIR CHARLES (d. 1648)

=2q; sin irv1(1-q2a) (1-2q2' 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 <m <101, but no complete proof of it ha:: See also:

• VET

62. Gauss's Sums.—Let m be any positive real integer; then S=--M —I =-m e2s2'n/m =i+i sin:. 1+i s=o This remarkable formula, when m is prime, contains results which were first obtained by Gauss, and thence known as Gauss's sums. The easiest method of proof is Kronecker's, which consists in finding the value off{e2dizi1mdz/(1-esniz)}, taken See also:

• ROUND (O. Fr. rond, Lat. rotundus, the Fr. is the source also of Du. rond; Ger., Swed., Dan. and Nor. rond)

Thus if n is an odd prime of the form 4k+l, E(-1);(8-1) =2, and the coefficient of q" is 8, which is right, because the one possible composition n = a2+b2 may be written n =(a)2+(b)2=(b)z+(a)2, giving eight representations. By methods of a similar character formulae can be found for the number of representations of a number as the sum of 4, 6, 8 squares respectively. The four-square theorem has been stated in § 41; the eight-square theorem is that the number of representations of a number as the sum of eight squares is sixteen times the sum of the cubes of its factors, if the given number is odd, while for an even number it is sixteen times the excess of the cubes of the even factors above the cubes of the odd factors. The five-square and seven-square theorems have not been derived from q-series, but from the general theory of quadratic forms. 68. Still more remarkable results are deducible from the theory of the transformation of the theta functions. The elementary formulae are On (u, w+I) =e" 4011(u, w), 010(u, w--I) =e'"`/4O1o(u, w), 001(u, w+I) =Ooo(u, w), Ooo(u, w+ I) =Bo1(u, co), e-"iu2/wO11 (u, co), w C 'tn2/welo (u, —j-) = -iwt3lo(n, w), w e'"`u2/wo'0 (u, —L) =J —iweol(u, w), w e-40/w800 /u, I =1/ Boo(u, w), ( w where —iw is to be taken in such a way that its real part is positive. Taking the definition of s given in § 67, and considering K as a function of w, we find K(w+I) ='Ielo218012 = 2K(w)/K (w), K(—1) =0012/0002 =K' (w). For convenience let K2(w)=a: then the substitutions (01,w-F1 ) and (w,—w-') convert a into a/(a—I) and (I —a) respectively. Now if a, 0, y, b are any real integers such that a5—07 =I, the substitution [w,(aw+R)/(yw+d)] can be compounded of (w, 07+1) and (w,—w-,); the effect on a will be the same as if we apply a corresponding substitution compounded of [a, a/(a—i)] and a, i —a]. But these are periodic and of order 3, 2 respectively] ; therefore we cannot get more than six values of a, namely a I a—I I a, I—a, — , I a—I~ I —aa 0 and any symmetrical function of these will have the same value at any two equivalent places in the modular dissection (§ 33). Their sum is See also:

• CONSTANT, BENJAMIN (1845-1902)

F. See also:

• KLEIN, JULIUS LEOPOLD (1810–1876)

Moreover, if with each form we associate its weight (§ 41) we find that with the notation of § 39 the degree of G is precisely EH(4n—K2)—e", where e„=1 when n is a square, and is zero in other cases. But this degree may be found in another way as follows. A complete representative set of trans-formations of order See also:

• RC(:N•OH)R'-RC(OH)

For instance ery "=884736743.9997775 . . •= 96o'+744 very nearly, and for the class (1, i, iI) the exact value of j is-96o3. 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 is-218, 3.53.233.293 71. The function j(w) is the most fundamental ni a set of quantities called class-invariants. 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—See also:

• FLY (formed on the root of the supposed original Tent. fleugan, to fly)

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+clxm-I ~- ... +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 them-selves: 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+x3-24.33.53=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 o-1) =72(w), whence it is found that (aw+$) (Yb+Ya-i ~s-Pa 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/ (j-1723), 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 See also:

• RESEARCH (O. Fr. recerche, from recercher, re- and cercer, mod. chercher, to search; Late Lat. circare, to go round in a circle, to explore)

73. In the Berliner Sitzungsberichte for the period 1883-189o, 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 See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

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)=x-I-P, 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—A<o rH(d) = Oh(—a) h(d) T+U,/d d>o H(d)=2Jdlogl—U,Id gm/2, J2(w)=2E(—I)i(m-I)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 left-hand 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 trans-formations, s-a 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 T2-DU2=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 See also:

• SIXTH

The actual calculation of the number of primes in a given See also:

• INTERVAL

Arithmetical Functions.-This term is applied to symbols such as 0(n), 4'(n), &c., which are associated with n by an See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)