Numbers of solutions of equations in finite fields

Such equations have an interesting history. In art. 358 of the Disquisitiones [1, a], Gauss determines the Gaussian sums (the so-called cyclotomic “periods”) of order 3, for a prime of the form p = 3n + 1, and at the same time obtains the number of solutions for all congruences ax− by ≡ 1 (mod p). He draws attention himself to the elegance of his method, as well as to its wide scope; it is only much later, however, viz. in his memoir on biquadratic residues [1, b], that he gave in print another application of the same method; there he treats the next higher case, finds the number of solutions of any congruence ax − by ≡ 1 (mod p), for a prime of the form p = 4n + 1, and derives from this the biquadratic character of 2 mod p, this being the ostensible purpose of the whole highly ingenious and intricate investigation. As an incidental consequence (“coronodis loco,” p. 89), he also gives in substance the number of solutions of any congruence y ≡ ax − b (mod p); this result includes as a special case the theorem stated as a conjecture (“observatio per inductionem facta gravissima”) in the last entry of his Tagebuch [1, c]; and it implies the truth of what has lately become known as the Riemann hypothesis, for the function–field defined by that equation over the prime field of p elements. Gauss’ procedure is wholly elementary, and makes no use of the Gaussian sums, since it is rather his purpose to apply it to the determination of such sums. If one tries to apply it to more general cases, however, calculations soon become unwieldy, and one realizes the necessity of inverting it by taking Gaussian sums as a starting point. The means for doing so were supplied, as early as 1827, by Jacobi, in a letter to Gauss [2, a] (cf. [2, b]). But Lebesgue, who in 1837 devoted two papers [3, a,b] to the case n0 = · · · = nr of equation (1), did not succeed in bringing out any striking result. The whole problem seems then to have been forgotten until Hardy and Littlewood found it necessary to obtain formulas for the number of solutions of the congruence ∑ i x n i ≡ b (mod p) in their work on the singular series for Waring’s problem [4]; they did so by means of Gaussian sums. More recently, Davenport and Hasse [5] have applied the same method to the case r = 2, b = 0 of equation (1) as well as to other similar equations; however, as they were chiefly concerned with other aspects of the problem, and in particular with its relation to the Riemann