MATHEMATICS: H. BLUMBERG 
375 
Moreover, our results have a most intimate connection with the work 
of every author cited, and, in most cases, the results bearing on the 
problem before us that are contained in the articles quoted subordinate 
themselves as special cases to the theorems obtained here.^ 
A sharp distinction has been made in the literature (as regards poly- 
nomials) between those investigations that are based on the divisibility 
properties of the coefficients and those that proceed from the consider- 
ation of the magnitude of the coefficients.^ One of the interesting results 
is that the gap between these two types of investigation may, in a cer- 
tain sense, be bridged. As a consequence, it is possible to show how 
the various theorems obtained flow from certain general considerations 
as a common source, and thus a surprising unification of the material 
is achieved.^ 
Our work deals with polynomials (for the various kinds of coefficients 
considered, see situations 1-5 inch), linear homogeneous differential 
expressions, linear homogeneous difference expressions (for the first 
time as regards our problem, as far as the writer knows), and more 
general expressions. There is no better way of making what is essential 
in the proofs come to the front than by treating the subject in abstract, 
postulational fashion. Thereby also the interconnection between the 
results and the unification above referred to are made manifest. Thus, 
it is easy to see, as Koenigsberger has pointed out by an example, that 
the Schoenemann-Eisenstein theorem cannot be directly extended to 
the case of linear homogeneous differential expressions; but our abstract 
treatment furthermore lays bare the underlying reason — by no means 
evident otherwise — why it breaks down, and at once indicates what 
analogous theorem may replace it.^ The abstract treatment is, more- 
over, especially fitting here because a small number of simple assumptions 
is sufficient for the foundations of the theory. 
We start^ with any aggregate @ whatsoever and shall deal with finite, 
ordered subaggregates E = (eo, ei, ^2, • • • , e^) oi ev [v = 0,1,2, - • , m] 
being an element of 
Such a finite, ordered subaggregate E we shall call a ^parenthesis^ of 
m will be called the 'order' of E. We assume that the 'product' 
A • B = {ao, ai, ^2, • • • , ai) • {bo, bi,b2,' • • , W of any two parentheses of 
® is equal to a parenthesis C = (co, Ci, c^, • ■ • , Cn) of © for which n = r-^ s. 
We assume furthermore that with every element e oi 'B there is asso- 
ciated a single number r], called the ' rank of e,' t] being an integer or 
— oo (never + oo), such that when A ^ B = C one of the following 3 sets 
of relations holds (see situations 1-10 that make either I or II or III 
valid) : 
