Discrete Fourier Transform DFT

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I'm having trouble locating a satisfactory reference on discrete Fourier transform (DFT). I would like to get a clear understanding for 1. DFT over $\mathbb{C}$ and 2. DFT over a finite field $\mathbb{F}_q$. My goal is to understand why these transforms are both called 'DFT.' I'm assuming because they satisfy very similar properties. Any suggestions for references would be greatly appreciated!