Two-view correspondence learning has recently focused on considering the coherence and smoothness of the motion field between an image pair. Dominant schemes include controlling the complexity of the field function with regularization or smoothing the field with local filters, but the former suffers from heavy computational burden, and the latter fails to accommodate discontinuities in the case of large scene disparities. In this paper, inspired by Fourier expansion, we propose a novel network called DeMatch, which decomposes the motion field to retain its main ``low-frequency'' and smooth part. This achieves implicit regularization with lower computational cost and generates piecewise smoothness naturally. Specifically, we first decompose the rough motion field that is contaminated by false matches into several different sub-fields, which are highly smooth and contain the main energy of the original field. Then, with these smooth sub-fields, we recover a cleaner motion field from which correct motion vectors are subsequently derived. We also design a special masked decomposition strategy to further mitigate the negative influence of false matches. All the mentioned processes are finally implemented in a discrete and learnable manner, avoiding the difficulty of calculating real dense fields. Extensive experiments reveal that DeMatch outperforms state-of-the-art methods in multiple tasks and shows promising low computational usage and piecewise smoothness property. The code and trained models are publicly available at https://github.com/SuhZhang/DeMatch.