Chap. VIII] CuBIC CONGRUENCES. 253 
integral roots y^, 2/2, Vs, and p is a prime = 1 (mod 3), and B^O (mod p) . Set 
^Vi = yi+ry2-\-r%, ^V2 = yi+r^y2+ry3, r^+r+ 1=0 (mod p). 
The roots of u^-\-Cu—B^/27= (mod p) are Ui = ^i^, U2 = ^2^. After finding 
Vi from Vi^=Ui (mod p), we get V2= —B/{3vi), and determine the y's from 
'2yi=0 and the expressions for 3^1, Sv2. Thus 
2/i=yi+z;2, y2=r\+rv2, ys^rvi+r^ (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 7^2 ^3 
"^"^7 ' +(~^+^7 ' ^2' ^ ' ^^ (modp). 
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. Oltramare^^^ found the conditions that one or all of the roots of 
x^+Spx+2q=0 (mod fx) given by Cardan's formula become integral modulo 
fx, a prime. Set 
D = q^-\-p\ a=-q+VD, T=-q-VD, 
First, let /x be a prime 6n — 1 . If D is a quadratic residue of /x, there 
is a single rational root — 2g/(p+(r^"+T^"). If Z) is a quadratic non-residue 
of fx, there are three rational roots or no root according as the rational part 
M of the development of o-^"~^ by the binomial theorem satisfies or does not 
satisfy ilfp^+g^O (mod //) ; if also /x = 18m+ll and there are three rational 
roots, they are 
2M^, -^(M±iW-3i)), 
if (72"*+i = ikf 4-iVVD; with a like result when m = 18m+5. 
Next, let /x = 6n+l. If Z) is a quadratic non-residue of ^x, there is one 
rational root or none according as the rational part M of the development 
of o-^" is or is not such that 
(2M-l)2(ikf+l)=-2gVp3 (modM), 
and if a rational root exists it is 2q/ \ p {2M — 1 ) } . If Z) is a quadratic residue 
of IX, there are three rational roots or none according as cr^''= 1 (mod /x) 
or not. When there are three, they are given explicitly if ^t=18m-|-7 or 
18m + 13, while if // = 18m + l there are sub-cases treated only partially. 
G. T. Woronoj^^^ (or Voronoi) employed Galois imaginaries a -{-hi, where 
i^—N=0 (mod p) is irreducible, p being an odd prime, to treat the solution of 
x^—rx — s=0 (mod p). 
I'lJour. fiir Math., 45, 1853, 314-339. 
'"Integral algebraic numbers depending on a root of a cubic equation (in Russian), St. Peters- 
burg, 1894, Ch. I. Cf. Fortschritte Math., 25, 1893-4, 302-3. Cf. Voronoi."' 
