CHAP, viii] CUBIC CONGRUENCES. 253 integral roots 3/1, y2> 2/s> and p is a prime ss 1 (mod 3) , and B ^ 0 (mod p) . Set 301=01+^2+^8, 3^=2/1+^2/2+^3, r2+r+l=0 (mod p). The roots of u?+Cu —2?3/27= o (mod p) are HI = t^3, w2 = *>23- After finding »i from VI^HI (mod p), we get v2= — J?/(3i>i), and determine the y's from S&SEO and the expressions for 3^, 3v2. Thus 2/2=^+^2, 2/3=^1+^2 (mod p). Since by hypothesis the cubic congruence has three distinct integral roots, the quadratic has two distinct integral roots, whence p-l p-l £2 £3 ti* 3 =1, D 2 =1 (mod p), I)=T+27> (r» \ g---1 / r \v~l *~l -|-J>V 3 + V"^+I)V 3 S2? D 2 sl (mod p)- Conversely, if the last two conditions are satisfied, the cubic congruence has three distinct real roots provided p=l (mod 3), B^O (mod p). G. Oltramare131 found the conditions that one or all of the roots of x3+3px+2g=0 (mod n) given by Cardan's formula become integral modulo fjL, a prune. Set First, let ju be a prime 6n— 1. If D is a quadratic residue of /-t, there is a single rational root — 2#/(p+<r2n+r2n). If D is a quadratic non-residue of ju, there are three rational roots or no root according as the rational part M of the development of o-271"1 by the binomial theorem- satisfies or does not satisfy Mp2+^=0 (mod fj) ; if also M = 18m+ll and there are three rational roots, they are if o*"+l**M+NVD; with a like result when M = Next, let /x = 6n+l- If D is a quadratic non-residue of JJL, there is one rational root or none according as the rational part M of the development of o-2n is or is not such that -2$2/p3 (mod/z), and if a rational root exists it is 2q/ \ p (2M — 1 ) } . If D is a quadratic residue of JLI, there are three rational roots or none according as 0-2nE= 1 (mod /*) or not. When there are three, they are given explicitly if /x=18m+7 or 18m +13, while if M = 18m+l there are sub-cases treated only partially. G. T. Woronoj132 (or Vorono'i) employed Galois imaginaries a+fo", where i2— JV==0 (mod p) is irreducible, p being an odd prime, to treat the solution of #3— rx— s=0 (mod p). l»lJour. fur Math., 45, 1853, 314-339. "'Integral algebraic numbers depending on a root of a cubic equation (in Russian), St. Petersburg, 1894, Ch. I. Cf . Fortschritte Math., 25, 1893-4, 302-3. Cf . Voronoi.109