On extensions of difference fields and the resolvents of prime difference ideals
Richard M. Cohn · Proceedings of the American Mathematical Society · 1952
Introduction. In this note we prove an analogue for difference fields2 of the theorem that every finite algebraic extension of a field of characteristic 0 is a simple extension.We apply this analogue to the theory of the resolvent system of a reflexive prime difference ideal.Our method is essentially that used in M.D.P. to obtain a weaker theorem; but we make a more careful study of the situation which exists before the indeterminates X, of M.D.P. (corresponding to both the di and X¿ of this note) are specialized, in order to overcome the difficulty which arises because, even in polynomial rings over difference fields of characteristic 0, there exist prime difference ideals containing no linear polynomial of effective order zero and yet admitting no more than one solution in any extension of the coefficient field.3This study is contained in §4 below.2. Definitions and statement of the theorem.We call a difference field % periodic or aperiodic according to whether or not there exists an integer n, fixed for %, such that every element of % is equal to its nth transform.If % is an aperiodic subfield of a difference field ®, and P is any nonzero difference polynomial in the ring4 ®{yi, • • • , y*}, there exist elements jui, • • • , pn in % which do not annul P when substituted for yi, • • • , yn respectively.6