We found an exact expression for the Flory radius R (F) of Gaussian polymers placed in an external periodic field. This solution is expressed in terms of the two parameters eta and a that describe the reduced strength of an external field and the period of the field to the polymer gyration radius ratio, respectively. R (F) is found to be a decaying function of eta for any values of a . Provided that the gyration radius is of the order of the period of an external field or less, the ground-state (GS) approximation of the exact result for R (F) is shown to give qualitatively incorrect results. In addition to the "ground-state" contribution, the exact solution for R (F) contains an additional term that is overlooked by the GS approximation. This term gives rise to the fact that R (F) as a function of eta exhibits power law behavior (rather than exponential decay obtained from the GS result) once eta exceeds the threshold value eta(con) .