In this paper the topological methods introduced by Kaplansky and the theory of linear compactifications are used to prove a result classifying the maximal immediate extensions of a valued field. Results on the existence of a complete discrete rank n valued field of characteristic 0 with prescribed residue class field of characteristic p > 0 are discussed. By applying results of Endler and Ribenboim the existence of a valued field of characteristic 0 and having prescribed residue field of characteristic p > 0 when the value group has finite rank but need not be discrete is demonstrated.