MATHEMATICS: H. BLUMBERG 
381 
1 Gauss, Disquisitiones arithmeticae (1801) Art. 341; Kronecker, /. Math., Berlin, 29, 280 
(1845), 100, 79 (1887),/. Math.,Paris, 19, 177 (1854) and ser. 2, 1, 399 (1856); Schoenemann,/. 
Math., Berlin, 32, 100 (1846) and 40, 188 (1850); Eisenstein, /. Math., Berlin, 39, 160(1850); 
Dedekind,/. Math., Berlin, 54, 27 (1857); Floquet,^ww. Sci. Ec. norm., Paris, (1879), cited by 
Koenigsberger; Koenigsberger, /. Math., Berlin, 115, 53 (1895), 121, 320 (1900), Math. Ann., 
Leipzig, 53, 49 (1900); Netto, Math. Ann., 48, 81 (1897); Perron, Math. Ann., 60, 448 (1905), 
/. Math., Berlin, 132, 288 (1907); M. Bauer, /. Math., Berlin, 128, 87 and 298, (1905), 132, 
21 (1907), 134, 15 (1908); Dumas, /. Math., Paris, ser. 6, 2, 191 (1906). 
2 Thus the work on the subject before the publication of the Schoenemann-Eisenstein 
theorem is summarized and generalized by that theorem, which, as previously indicated, is 
a very special case of theorem IV. In fact, theorem I alone, for example, includes as special 
cases a great bulk of the results heretofore pubHshed, and, in particular, the following: the 
theorem of Floquet — quoted by Koenigsberger — , all the results contained in the 1895 paper 
of Koenigsberger, nearly all contained in his 1900 Math. Ann. paper, almost all the results 
of Netto and almost all the results in the 1905 paper of Perron. Cf. also the paragraph pre- 
ceding the statement of theorem V. 
^ Perron, /. Math., Berlin, 132, 288, and Loewy, in Pascal's Repertorium (1910) Analysis Ii, 
292 and 293. 
* I may perhaps be permitted to remark, for the purpose of indicating that the results 
given are less artificial than one would at first suppose, that I had obtained my chief results 
with little knowledge of the hterature. Their intimate connection with results already 
obtained points to a degree of 'naturalness' of these results that one would hardly attribute 
to them in the absence of such a connection. 
^ As a matter of fact, two distinct theorems obtained may be properly regarded as (highly) 
generalized Schoenemann-Eisenstein theorems for the case of linear homogeneous differential 
expressions: theorem I for situation 8 and theorem XII for situation 10. Curiously, Koenigs- 
berger himself has obtained theorems — special cases of XII — for differential expressions 
that may be properly regarded as generalized Schoenemann-Eisenstein theorems, without 
his having noticed this relation. 
^ It is possible to build up just as general a theory as ours by dealing exclusively with 
* parentheses' whose elements are 'ranks' — see below in the same paragraph — and hence 
always integers or— oo . The reader who prefers a more concrete, tho necessarily less gen- 
eral, discussion may, for example, at once interpret ©, 'parenthesis,' 'product' and 'rank' — 
see below in the same paragraph — as in situation 8. In that case, II will hold. 
' In general, we denote the rank of an element represented by a Latin letter by the 
corresponding Greek letter. By ^^^^^^ (a:x + /3^), x+^^u x+J!^^^a = v 
(ax-r/S/i+o") — see II and III below — we naturally understand the largest value attained 
by the numbers of the set ax + jS/z, a\ + ^u, (X\ -{- -\- a, X, fx and <r varying so as 
to satisfy the relations iJ, = v, \-{- fi^P, fX-}- (T =v, and in addition, of course,. 
0 ^ X ^ r and 0 ^ /i ^ s. 
8 Hensel, /. Math., Berlin, 127, 51-84, §2 (1904) or Zahlentheorie (1913), chaps. 3 and 6. 
* More generally, the numerical coefiQcients may belong to any abstract system {K, X), 
where K is & class, such that if a and b are elements of K both a b and a X 6 are elements 
of K. This remark applies just as well to the numerical coefficients in situations 4-10 incl. 
The question of convergence does not enter here because the formal character of the 
series is sufficient for our purpose. More generally, we may have such series in two or more 
variables, the required change in the definition of rank being evident. 
'1 These lemmas lead up to the theorems I-X and are given partly for the purpose of 
indicating the nature of the proofs and partly because they are believed to be of interest 
in themselves. 
