numbers. He also succeeded in showing that in the field R(e^2πi/p>) the equation α^p+β^p+γ^p=0 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^m+y^m=z^m has no rational solutions, except the obvious ones for which xyz=0. 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 has yet been obtained.

Hilbert has studied in considerable detail what he calls Kummer fields, which are obtained by taking x, a primitive pth root of unity, and y any root of yi'-a=o, where a is any number in the field R(x) which is not a perfect pth power in that field. The Kummer field is then R (x, y), Consisting of) all rational functions of x and y. ()ther fields that have been discussed more or less are general cubic fields, some special bi quadratic and a few Abelian fields not cyclic. Among the applications of cyclotomy may be mentioned the proof which it affords of the theorem, first proved by Dirichlet, that if m, n are any two rational integers prime to each other, the linear form mx +n is capable of representing an infinite number of rimes.

62. Gauss's Sums.-Let m be any positive real integer; then S=m'I

i+i=-»»

QZSQFI/m=1 m

I-l-I V

r=o

This remarkable formula, when m is prime, contains results which were first obtained be' Gauss, and thence known as (§ auss's sums. The easiest method o proof is Kronecker's, which consists in finding the value of f {e21fi2”"'dz/(I~e2"i2)}, taken round an appropriate contour. It will be noticed that one result of the formula is that the square root of any integer can be expressed as a rational function of roots of unity.

The most important application of the formula is the deduction from it of the law of quadratic reciprocity for real primes: this was done by Gauss. .

63. One example may be given of some remarkable formulae giving explicit solutions of representations of numbers by certain quadratic forms. Let p be any odd prime of the form 7n+2; then we shall have p=7n+2=x2+7y2, where x is determined by the congruences (),

n .

2x~ (m0d P), x-3 (mod 7)-This

formula was obtained by Eisenstein, who proved it by investigating properties of integers in the field generated by 'l]a'2I1]"7=0| which is a component of the field generated b seventh roots- of unity. The first formula of this kind was given by gauss, and relates to the case p=4n+I =x' -|-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), ¢»(x) are polynomials in x with rational integral coefficients. The congruence f(x)E¢(x) (mod p) is identical when each coefficient of f is congruent, mod p, to the corresponding coefficient of ¢. It will be convenient to write, under these circumstances, f~¢(mod p) and to say that f, ¢ are equivalent, mod p. Every polynomial 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, pf' 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~cf1f¢ ...fl where c is an integer and f, , ff2, ...f1 are prime functions; this resolution is unique. Moreover, it dollows from Fermat's theorem that {F(x)}P~F(xP), {F(x)}1'2..F(xP2), an soon

As in the case of equations, it may be proved that, when the modulus is prime, a congruence f(x)-E o (mod p) cannot have more in congruent roots than the index of the highest power of x in f(x), and that if xE£is a solution, f(x)-(x-£)f1(x), where f1(x) is another polynomial. The solutions of xPEx are all the residues ofp; hence xP~x-x(x+1) (x-+-2) . . .(x-l-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(x"”"'-I)~Uf(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) I %(Pm 2pm/¢+EPm/ab )

where a, b, c are the different prime factors of m. Moreover, if F is any given function, we can find a resolution F~cF1F2 . . . F, ,, (mod p)

where c is numerical, F1 is the product. of all prime linear functions which divide F, F 2 is the product of all the prime quadratic factors, and so ont

65. By the functional congruence ¢(x)2»//(vc) (mod p, f(x)) is meant that polynomials U, V can be found such that 4>(x) ===, Z/(x)+pU+ Vf(x) identically. We might also writeq>(x)~/»(x) (mod p, f(x)); but this is not so 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 ¢(x), there will be definite functions X(x), ¢, (x) such that 4>(x) =f(x)X(x) -l-q'>i(x) where ¢1(x) is of lower dimension than f(x); moreover, we may suppose qi-1(x) replaced by the equivalent reduced function 4>2(x) mod p. Finally then, ¢E4>2 (mod p, f(x)) where ¢2 is a reduced function, mod Q, of order not greater than (m-I). If we put pm =n, there will be in all (including zero) n residues to the compound modulus (p, f): let us denote these by Ri, R2, . . . R, ,. Then (cf. § 28) if we reject the one zero residue (Rn, suppose) and take any function¢ of which the residue is not zero, the residues of ¢»R1, 4>R2, . . ¢>R, , 1 will all be different, and we conclude that 4>""E I (mod 1), f). Every function therefore satisfies ¢"~¢> (mgd 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 fact that there are integral polynomials M, X', Y', Z', T' such that U =mM +X'X +Y'Y-I-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.

t is worth mentioning that one of Gauss's proofs of the law of quadratic reciprocity (Gétt. Naehr. 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 orms of the third and higher degree is still guite fragmentary. . jordan has proved that the class number is nite. . Poincaré has discussed the classification of ternary and qliiaternary cubics. With regard to the ternary cubic it is known t at from any rational solution of f =o we can deduce another by a process whici is equivalent to finding the tangential of a point (x1, y1, zr) on the curve, that is, the point where the tangent at (xt, yt, zi) meets the curve again. We thus obtain a series of solutions (xl, yl, zt), (xg, y2, zz), &c., which may or may not be periodic. E. Lucas and J. Sylvester have proved that for certain cubics f =o has no rationa solutions; for instance x3-I-y3-Az3=o has rational solutions only if A=ab(a-|-b)/ca, 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 I9 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 lacobian theta functions. Let w be any complex quantity such that the real part of iw is negative; and let g =e"i~. Then A +ou

000(1J)=2qs2e2°"'i" = I -l-2g cos 21rv -I-2q4 cos 41r°v-I-2gP cos 61rv+ . . . oo

0°

=II(I-q'“)(1 +2Q2"1 COS 2r'v+q'~"“).

I

001('u) = I -2g cos 21rv-I-2g4 cos 41m -zqp cos 6-rrv+ . . . ='€I(I -q”)(I -2q”"' Cos 21rv+q*““).

6w(v) =2ql cos 1|-11+ zqf cos 31rv+2qi" cos 51rv+ . . =2qi cos 1rv°ff(1 -q2”)(1 +2q2' cos 2-:rv-l-q4=°), , 0n(v) =2q} sin -/rv-zqi sin 31r-v -I-2q”'° sin 51r-o-. . r =2q& sin 7l"U§ (I -g2°) (I-2q2-' cos 21rv+q4'). Instead of 0w(o), &c., we write 600, &c. Clearly 011=o; we have the important identities

611, = 7600010001 300' '= 0014 +9104

where 911' means the value of d0n(v)/dv for v =o. If, now, we put ~/x=g-£1 /K' =%:» 'l4=7l'0002U,