  We show how the quantum fast Fourier transform (QFFT) can be made exact for
arbitrary orders (first for large primes). For most quantum algorithms only the
quantum Fourier transform of order $2^n$ is needed, and this can be done
exactly. Kitaev \cite{kitaev} showed how to approximate the Fourier transform
for any order. Here we show how his construction can be made exact by using the
technique known as ``amplitude amplification''. Although unlikely to be of any
practical use, this construction e.g. allows to make Shor's discrete logarithm
quantum algorithm exact. Thus we have the first example of an exact non black
box fast quantum algorithm, thereby giving more evidence that ``quantum'' need
not be probabilistic. We also show that in a certain sense the family of
circuits for the exact QFFT is uniform. Namely the parameters of the gates can
be calculated efficiently.
