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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 𝑦^𝑝－𝑎＝0, 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. Other fields that have been discussed more or less are general cubic fields, some special biquadratic 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 primes.

62. Gauss’s Sums.—Let 𝑚 be any positive real integer; then

$\sum _{s=0}^{s=m-1}e^{2s^{2}\pi i/m}={\frac {i+i(-i)^{m}}{1+i}}{\sqrt {m}}$

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 of ∫{𝑒^{2𝜋𝑖𝑧²/𝑚}𝑑𝑧/(1－𝑒^{2𝜋𝑖𝑧})}, 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

2𝑥≡(3𝑛)!/(𝑛)! (2𝑛)!(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 𝜂³－21𝜂－7＝0, which is a component of the field generated by seventh roots of unity. The first formula of this kind was given by Gauss, and relates to the case 𝑝＝4𝑛＋1＝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), 𝜙(𝑥) are polynomials in 𝑥 with rational integral coefficients. The congruence 𝑓(𝑥)≡𝜙(𝑥) (mod 𝑝) is identical when each coefficient of 𝑓 is congruent, mod 𝑝, to the corresponding coefficient of 𝜙. It will be convenient to write, under these circumstances, 𝑓∼𝜙(mod 𝑝) and to say that 𝑓, 𝜙 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, 𝑝^ℎ 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∼𝑐𝑓₁𝑓₂ . . . 𝑓_𝑙 where 𝑐 is an integer and 𝑓₁, 𝑓₂,...𝑓_𝑙 are prime functions; this resolution is unique. Moreover, it follows from Fermat’s theorem that {𝖥(𝑥)}^𝑝∼𝖥(𝑥^𝑝),{𝖥(𝑥)}^𝑝²∼𝖥(𝑥^𝑝²), and so on.

As in the case of equations, it may be proved that, when the modulus is prime, a congruence 𝑓(𝑥)≡0 (mod 𝑝) cannot have more in congruent roots than the index of the highest power of 𝑥 in 𝑓(𝑥), and that if 𝑥≡𝜉 is a solution, 𝑓(𝑥)∼(𝑥－𝜉)𝑓₁(𝑥), where 𝑓₁(𝑥) is another polynomial. The solutions of 𝑥^𝑝≡𝑥 are all the residues of 𝑝; hence 𝑥^𝑝－𝑥∼𝑥(𝑥＋1)(𝑥＋2) . . .(𝑥＋𝑝－1), 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^𝑝𝑚－1－1)∼𝚷𝑓(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＞1)

1/𝑚(𝑝^𝑚－𝚺𝑝^𝑚/𝑎＋𝚺𝑝^{𝑚/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_𝑚(mod 𝑝)

where 𝑐 is numerical, 𝖥₁ is the product of all prime linear functions which divide 𝖥, 𝖥₂ is the product of all the prime quadratic factors, and so on.

65. By the functional congruence 𝜙(𝑥)≡𝜓(𝑥) (mod 𝑝,𝑓(𝑥)) is meant that polynomials 𝖴, 𝖵 can be found such that 𝜙(𝑥)＝𝜓(𝑥)＋𝑝𝖴＋𝖵𝑓(𝑥) identically. We might also write 𝜙(𝑥)∼𝜓(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 of 𝑥^𝑚 is unity, and it will be further assumed that 𝑓(𝑥) is a prime function, mod 𝑝. Whatever the dimensions of 𝜙(𝑥), there will be definite functions 𝜒(𝑥), 𝜙₁(𝑥) such that 𝜙(𝑥)＝𝑓(𝑥)𝜒(𝑥)＋𝜙₁(𝑥) where 𝜙₁(𝑥) is of lower dimension than 𝑓(𝑥); moreover, we may suppose 𝜙₁(𝑥) replaced by the equivalent reduced function 𝜙₂(𝑥) mod 𝑝. Finally then, 𝜙≡𝜙₂ (mod 𝑝, 𝑓(𝑥)) where 𝜙₂ is a reduced function, mod 𝑝, of order not greater than (𝑚－1). If we put 𝑝^𝑚＝𝑛, there will be in all (including zero) 𝑛 residues to the compound modulus (𝑝, 𝑓): let us denote these by 𝖱₁, 𝖱₂, . . . 𝖱_𝑛. Then (cf. § 28) if we reject the one zero residue (𝖱_𝑛, suppose) and take any function 𝜙 of which the residue is not zero, the residues of 𝜙𝖱₁, 𝜙𝖱₂, . . . 𝜙𝖱_𝑛－1 will all be different, and we conclude that 𝜙^𝑛－1≡1 (mod 𝑝, 𝑓). Every function therefore satisfies 𝜙^𝑛∼𝜙 (mod 𝑝, 𝑓); by putting 𝜙＝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∼0 (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＋𝖹′𝖹＋𝖳′𝖳

identically. In this notation 𝖴∼𝖵 means that 𝖴－𝖵∼0. 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 (Gött. Nachr. 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. Poincaré 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＝0 we can deduce another by a process which is equivalent to finding the tangential of a point (𝑥₁, 𝑦₁, 𝑧₁) on the curve, that is, the point where the tangent at (𝑥₁, 𝑦₁, 𝑧₁) meets the curve again. We thus obtain a series of solutions (𝑥₁, 𝑦₁, 𝑧₁), (𝑥₂, 𝑦₂, 𝑧₂), &c., which may or may not be periodic. E. Lucas and J. J. Sylvester have proved that for certain cubics 𝑓＝0 has no rational solutions; for instance 𝑥³＋𝑦³－𝖠𝑧³＝0 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 𝜔 be any complex quantity such that the real part of 𝑖𝜔 is negative; and let 𝑞＝𝑒^𝜋𝑖𝜔. Then

$\theta _{00}(v)=\sum _{-\infty }^{\infty }1+q^{s^{2}}e^{2s\pi iv}=1+2q\cos 2\pi v+2q^{4}\cos 4\pi v+2q^{9}\cos 6\pi v+...$

$\qquad \quad =\prod _{1}^{\infty }(1-q^{2s})(1+2q^{2s-1}\cos 2\pi v+q^{4s-2}),$

$\theta _{01}(v)=1-2q\cos 2\pi v+2q^{4}\cos 4\pi v-2q^{9}\cos 6\pi v+...$

$\qquad \quad =\prod _{1}^{\infty }(1-q^{2s})(1-2q^{2s-1}\cos 2\pi v+q^{4s-2}),$

$\theta _{10}(v)=2q^{1 \over 4}\cos \pi v+2q^{9 \over 4}\cos 3\pi v+2q^{25 \over 4}\cos 5\pi v+...$

$\qquad \quad =2q^{1 \over 4}\cos \pi v\prod _{1}^{\infty }(1-q^{2s})(1+2q^{2s}\cos 2\pi v+q^{4s})$

$\theta _{11}(v)=2q^{1 \over 4}\sin \pi v-2q^{9 \over 4}\sin 3\pi v+2q^{25 \over 4}\sin 5\pi v-...$

$\qquad \quad =2q^{1 \over 4}\sin \pi v\prod _{1}^{\infty }(1-q^{2s})(1-2q^{2s}\cos 2\pi v+q^{4s})$

Instead of 𝜃₀₀(0), &c., we write 𝜃₀₀, &c. Clearly 𝜃₁₁＝0; we have the important identities

𝜃₁₁′＝𝜋𝜃₀₀𝜃₁₀𝜃₀₁    𝜃₀₀⁴＝𝜃₀₁⁴＋𝜃₁₀⁴

where 𝜃₁₁′ means the value of 𝑑𝜃₁₁(𝑣)/𝑑𝑣 for 𝑣＝0. If, now, we put

${\sqrt {\kappa }}={\theta _{10} \over \theta _{00}},\quad {\sqrt {\kappa '}}={\theta _{01} \over \theta _{00}},\quad u=\pi \theta _{00}{}^{2}v$