Inversive difference fields
Richard M. Cohn · Bulletin of the American Mathematical Society · 1949
INVERSIVE DIFFERENCE FIELDS 597 illustration of this phenomenon we give the following example of an ideal with a basis which is a subset of its basic set. 10 Let J be the field of all rational functions of x with transforming defined as the operation of replacing x by x + 1.We consider the polynomials (2) yi -y, z 2 -y, zi -z.Using Theorem IX of M.D.P. one shows easily that (2) is a basic set of a prime reflexive ideal A with coefficients in J. Since the initials of the polynomials of ( 2) are unity, it follows that it is a basis for A.But the equations z\ -JS = 0, z 2 -y = 0, imply that yi = zl=z 2 =y; so that any solution of these equations is also a solution of yi -y = 0. Thus z 2 -y, z\ -z is itself a basis for A.