State-of-the-art word embeddings presume a linear vector space, but this approach does not easily incorporate the nonlinearity that is necessary to represent polysemy. We thus propose a novel semantic FIeld REepresentation, called FIRE, which is a $D$-dimensional field in which every word is represented as a set of its locations and a nonlinear function covering the field. The strength of a word's relation to another word at a certain location is measured as the function value at that location. With FIRE, compositionality is represented via functional additivity, whereas polysemy is represented via the set of points and the function's multimodality. By implementing FIRE for English and comparing it with previous representation methods via word and sentence similarity tasks, we show that FIRE produces comparable or even better results. In an evaluation of polysemy to predict the number of word senses, FIRE greatly outperformed BERT and Word2vec, providing evidence of how FIRE represents polysemy. The code is available at https://github.com/kduxin/firelang.