In this work, we explore how undergraduate students use classical field theory when describing physical phenomena in the context of introductory electromagnetism. We have extracted five key characteristics of the electric and magnetic field from a historical analysis of the topic. These characteristics informed the creation of a questionnaire comprising six free-response conceptual questions. The questionnaire instrument was administered to undergraduate students in three European countries. Phenomenographical analysis of the students' responses shows that many undergraduates do not have a coherent idea of field theory. We conclude that, rather than focusing on teaching rules with which to calculate, more attention should be paid to the specific characteristics of field theory and the difference between fields and forces, with particular emphasis on the conceptual interpretation of the interaction process.