254 HISTORY OP THE THEORY OF NUMBERS. (CHAP, vm If 4r3 27s2 is a quadratic non-residue of p, the congruence has one and only one root; but if it is a residue, there are three roots or no root. G. Cordone133 gave simpler proofs of Oltramare's131 theorem II on the case jit = 6n 1, gave theorems to replace VII and VIII, and proved that the condition in IX is sufficient as well as necessary. Ivar Damm9* found when Cardan's formula gives three real roots, one or no real root of a cubic congruence, and expressed the roots by use of his quasi sine and cosine functions. For the prime modulus p = 3n+l, f = x3 + ax + b is irreducible if If p=3n 1, it is irreducible if c and ( b/2+c)n are both imaginary. There are given (p. 52) explicit expressions for b such that / is irreducible. J. Iwanow134 gave another proof of the theorem of Woronoj.132 Woronoj135 gave another proof of the same theorem and stated that the congruence has the same number of roots for all primes represen table by a binary quadratic form whose determinant equals 4r3+27s2. G. Arnoux136 gave double-entry tables of the roots of the congruences xr+bx* -fa=0 (mod ra), and solved numerical cubic congruences by interpreting Cardan's formulas. G. Arnoux137 treated x34-bx+a=0 (mod m) by use of Cardan's formula. For m = 11, he gave a table of the real roots for a^ 10, fc^j 10, and the residues of When R is a quadratic residue, the cube roots of a/2=±= VJR are found by use of a table for the Galois field of order II2 defined by i2^2 (mod 11), and the cubic is seen to have a real and two imaginary roots involving i. If R is a quadratic non-residue, there are three real roots or none. Like results are said to hold when m 1 is not divisible by 3. If ra= 1 (mod 3), there is a single real root if R is a quadratic non-residue; three real or three imaginary roots of the third order if J? is a residue. L. E. Dickson138 proved that, if p is a prime >3, x3+j3x-ffe=0 (mod p) has no integral root if and only if 4/33 27fe2 is a quadratic residue of pf say =s81/*2, and if §( 6+AtV 3) is not congruent to the cube of any y+z'V 3, where y and z are integers. The reducible and irreducible cubic congruences are given explicitly. Necessary and sufficient conditions for the irreducibility of a quartic congruence are proved. mRendiconti Circolo Mat. di Palermo, 9, 1895, 221-36. 1MBull. Ac. Sc. St. Petersburg, 5, 1896, 137-142 (in Eussian). 136Natural Sc. (Russian), 10, 1898, 329; cf. Fortschritte Math., 29, 1898, 156. 1MAssoc. franc.. av. sc., 30, 1901, II, 31-50, 51-73; corrections, 31, 1902, II, 202. »7Assoc. franc, av. sc.. 33, 1904, 19^-230 [182-1991. and Arnoux112. 166-202.