A new fast algorithm for the Fourier transforms over any finite field is introduced. It is highly regular and easy to be implemented by a pipelined structure

requiring only O （n log2n） additions and the same number of multiplications. In some fields

the number of multiplications can be reduced to O （n logn）. A flow chart and a schematic diagram for the hardware implementation are shown. The computational complexity and the throughput rate are estimated. It has applications in encoding and decoding of some error control codes in the spectral mode. It is also a fast algorithm for the evaluation of a polynomial over a finite field.