  Let k be a field. A "field generator" is a polynomial F in k[X,Y] satisfying
k(F,G) = k(X,Y) for some G in k(X,Y). If G can be chosen in k[X,Y], we call F a
"good field generator"; otherwise, F is a "bad field generator". These notions
were first studied by Abhyankar, Jan and Russell in the 1970s. The present
paper introduces and studies the notions of "very good" and "very bad" field
generators. We give theoretical results as well as new examples of bad and very
bad field generators.
