92 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



has distinct roots not divisible by p. Such a system of linear congruences 

 has been discussed by W. Burnside. 250 



E. Steinitz 251 stated that all theorems oh linear congruences follow 

 easily from one: Given k linear congruences in n variables modulo m, the 

 k sets of coefficients form the basis of a Dedekind Modul. If e\, , e n 

 are the invariants of this Modul (the last n r e's are zero if the rank 

 r is < ri) and if [e,-, m] is the g.c.d. of e and m, then the totality of sets of 

 solutions of the k congruences represent a Modul with the invariants 



7?i m 



\JB n , 



Expositions in the texts by Bachmann, 219 J. Konig, 221 and Cahen 223 have 

 been cited. Zsigmondy 230 found the number of solutions of a system of two 

 special congruences. 



H. Weber 252 made a direct examination of the conditions under which 



(2) fliyt/i + 02,2/2 -f + a pj y p = (mod p*) (j = 1, , /*) 



shall require that each yi be divisible by p", where p is a prime. It is as- 

 sumed that not every a# is divisible by p (otherwise a solution is obtained 

 by taking each y f to be any multiple of p"" 1 )- We may assume that 

 A = a>ij\i, /=i, ..., T is not divisible by p, while every (T -f- l)-rowed deter- 

 minant of the matrix (a,-/) is divisible by p. Denote the signed minors of 

 A by AM and set 



Thus Dfe, = A if fc = s; >* = if s ^ r, s =f= &; while, if s > r, D ks is a T- 

 rowed determinant of (a,-y). Applying Cramer's rule to the first r con- 

 gruences (2), we get 



(3) Ayj + D Jtr+l y r+l + + D, p y p = (mod p") (j = 1, -, r}. 



Hence 



A(ai#i + + a pr y p ) = A T+1>r y r+1 + + A pr y p (mod p"), 

 where 



equals a (r + l)-rowed determinant of (a t -,-) and hence is divisible by p. 

 Thus, if T < p, (2) are satisfied when y r+l , , y p are divisible by p ff " 

 In order that (2) shall require that each y t be divisible by p" it is therefore 

 necessary that T = p. This condition is also sufficient, since (3) then 

 reduce to AS/I = 0, , Ay p = 0, whence ?/i, , y f are divisible by p w . 

 F. Riesz 253 stated that, if the a ik and fa are real, the congruences 



= fa (mod 1) (i = 1, , m) 



k=l 



260 Messenger Math., 24, 1894, 51. 



261 Jahresbericht d. Deutschcn Math.-Verein., 5, 1896 [1901], 87. 



2 " Lehrbuch der Algebra, 2, 1896, 87-8; ed. 2, 1899, 94. Cf. Smith. 242 

 253 Comptes Rendus Paris, 139, 1904, 459-462. 



