Transcendental entire functions mapping every algebraic number field into itself

T. Schneider [1] has shown that a transcendental function with a limited rate of growth cannot assume algebraic values at too many algebraic points. It is not clear however whether a transcendental function may assume algebraic values at all algebraic points in an open set in its domain of analyticity. We show, using a method of B. H. Neumann and R. Rado [2] that this can be the case. Indeed we construct transcendental entire functions which, together with all their derivatives, assume, at every point in every algebraic number field, values in that field.