226 History of the Theory of Numbers. [Chap, viii 
G. Raussnitz*^ proved the theorem, due to Konig: Let 
(2) f{x) =aoX^-2+OiX''-H . . . +ap_2, 
where the a's are integers and ap_2 is not divisible by the prime p. Then 
f{x)=0 (mod p) has real roots if and only if the cyclic determinant 
(3) D = 
Oo Oi 02 ... ap_3 ap_2 
«! a2 03 ... ap_2 Oo 
Op_20o Oi ... Op_4 Op.s 
is divisible by p. In order that it have at least k distinct real roots it is 
necessary that all p — k rowed minors of D be divisible by p. If also not 
all p — k — 1 rowed minors are divisible by p, the congruence has exactly 
k distinct real roots. 
The theorem is applicable to any congruence not ha\'ing the root zero, 
since we may then reduce the degree to p — 2 by Fermat's theorem. 
Gustav Rados-'* proved Konig's theorem, using the fact that a system of 
p — l linear homogeneous congruences modulo p in p — 1 unknowns has at 
least k sets of solutions linearly independent modulo p if and only if the 
p — k rowed minors are divisible by p. 
L. Kronecker^^ noted that, if p is a prime, the condition for the existence 
of exactly p—m — 1 roots of (2), distinct from one another and from zero, 
is that the rank of the system 
(3') (a,+,) (i,A: = 0, l,...,p-2) 
modulo p is exactly m, where Os+p_i = a^. The same is the condition for 
the existence of a {p—m — l)-fold manifold of sets of solutions of the system 
of linear congruences 
2'a,+,0,= (mod p) ( A = 0, 1 , . . . , p - 2) . 
fc=0 
L. Kronecker^^ gave a detailed proof of his preceding results, noted that 
the rank is ?« if not all principal t?? -rowed minors are divisible by p while 
all VI -{-1 rowed minors are, and added that Co+Ci.t+ . . . +Cp_2a:^~^=0 
(mod p) has exactly s roots 7^0 if one and the same linear homogeneous 
congruence holds between every set of p—s (but not fewer) successive 
terms of the periodic series Cq, Ci, . . . , Cp_2, Cq, Ci, . . . . 
L. Gegenbauer^^ proved Kronecker's version of Konig's theorem. 
Gegenbauer^^ noted that Kronecker's theorems imply the corollary: 
"Math, und Naturw. Berichte aus Ungam, 1, 1882-3, 266-75. 
"Jour, fiir Math., 99, 1886, 258-60; Math. Termea Ertesito, Magj'ar Tudon Ak., Budapest, 1, 
1883, 296; 3, 1885, 178. 
»Jour. fur Math., 99, 1886, 363, 366. 
**Vorlesungen liber Zahlentheorie, 1, 1901, 389-415, including several additions by Hensel 
(pp. 393, 399, 402-3). 
"Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 165-9, 610-2. 
*^Ibid., 98, Ila, 1889, p. 32, foot-note. Cf. Gegenbauer.« 
