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Full text of "History Of The Theory Of Numbers - I"

226                    HISTORY OF THE THEORY OF NUMBERS.          [CHAP, vm
G. Raussnitz23 proved the theorem, due to Konig:   Let (2)                          /(a?) =o<^2+aiXp-8+... +ap_2,
where the a's are integers and ap_2 is not divisible by the prime p.   Then /(z)=0 (mod p) has real roots if and only if the cyclic determinant
a2
(3)
i    ...    ap_4   ap_3
is divisible by p. In order that it have at least k distinct real roots it is necessary that all pA; rowed minors of D be divisible by p. If also not all pk  1 rowed minors are divisible by p, the congruence has exactly k distinct real roots.
The theorem is applicable to any congruence not having the root zero, since we may then reduce the degree to p2 by Fermat's theorem.
Gustav Rados24 proved Konig's theorem, using the fact that a system of p1 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. Kronecker25 noted that, if p is a prune, the condition for the existence of exactly pml roots of (2), distinct from one another and from zero, is that the rank of the system
(3')                             (ai+k)           (i, k = 0, 1,..., p - 2)
modulo p is exactly m, where as+p_i=aa. The same is the condition for the existence of a (pwl)-fold manifold of sets of solutions of the system of linear congruences
2 cLji+i,<t>k=Q (mod p)             (/i = 0, 1,. .., p  2).
L. Kronecker26 gave a detailed proof of his preceding results, noted that the rank is m if not all principal m-rowed minors are divisible by p while all nt+l rowed minors are, and added that Co+CxX-l-.. .+cp_2zp~2=0 (mod p) has exactly s roots 5^0 if one and the same linear homogeneous congruence holds between every set of ps (but not fewer) successive terms of the periodic series CQ, clt..., cp_2, c0, cl}....
L. Gegenbauer27 proved Kronecker's version of Konig's theorem.
Gegenbauer28 noted that Kronecker's theorems imply the corollary:
^Math. und Naturw. Berichte aus Ungarn, 1, 1882-3, 266-75.
24Jour, fiir Math., 99, 1886, 258-60; Math. Termes Ertesito, Magyar Tudon Ak., Budapest, 1,
1883, 296; 3, 1885, 178. "Jour, fiir Math., 99, 1886, 363, 366. 26Vorlesungen iiber Zahlentheorie, 1, 1901, 389-415, including several additions by Hensel
(pp. 393, 399, 402-3).
27Sitzungsber. Ak. Wiss. Wien (Math.), 95, II, 1887, 165-9, 610-2. 28/6id., 98, Ila, 1889, p. 32, foot-note.   Cf. Gegenbauer.16