THE THEORY OF CONGRUENCES 815 25,sec.4).Euler had given a complete statement much like Gauss's in one paper of his Opuscula Analytica of 783 (Chap.25,sec.4).Nevertheless in article 151 of his Disquisitiones Gauss says that no one had presented the theorem in as simple a form as he had.He refers to other work of Euler including another paper in the Opuscula and to Legendre's work of 1785 Of these papers Gauss says rightly that the proofs were incomplete. Gauss is supposed to have discovered a proof of the law in 1796 when he was nineteen.He gave another proofin the Disquisitiones and later published four others.Among his unpublished papers two more were found.Gauss says that he sought many proofs because he wished to find one that could be used to establish the biquadratic reciprocity theorem (see below).The law of quadratic reciprocity,which Gauss called the gem of arithmetic,is a basic result on congruences.After Gauss gave his proofs,more than fifty others were given by later mathematicians Gauss also treated congruences of polynomials.If 4 and B are two polynomials in x with,say,real coefficients then one knows that one can find unique polynomials Q and R such that A=B.Q+R, where the degree of R is less than the degree of B.One can then say that two polynomials A and A2 are congruent modulo a third polynomial P if they have the same remainder R on division by P. Cauchy used this idea3 to define complex numbers by polynomial congruences.Thus if f(x)is a polynomial with real coefficients then under division by x+1 f(x)≡a+bx modx2+1 because the remainder is of lower degree than the divisor.Here a and b are necessarily real by virtue of the division process.If g(x)is another such polynomial then g(x≡c+dx mod2+1. Cauchy now points out that if A1,A2,and B are any polynomials and if A1 BQ1 R1 and A2 BQ2 R2 then A1+A2=R1 R2 mod B,and A142=RR2 mod B. We can now see readily that f(x)+g(x)(a +c)+(b +d)x mod x2+1 3.Exercices d'analyse et de physique mathimatique,4,1847,84.=Eupres,(1,10,312-23 and (2),14,93-120