For a given selection of rows and columns from a Fourier matrix, we give a number of tests for whether the resulting submatrix is Hadamard based on the primitive sets of those rows and columns. In particular, we demonstrate that whether a given selection of rows and columns of a Fourier matrix forms a Hadamard submatrix is exactly determined by whether the primitive sets of those rows and columns are compatible with respect to the size of the Fourier matrix... This motivates the creation of compatibility graphs for the Fourier matrices. We conclude with some results that facilitate the construction of these graphs for submatrix sizes 2 and 3. (read more)