Cooley-Tukey FFT over $\mathbb{Q}_p$ via Unramified Cyclotomic Extension

The reason why Cooley-Tukey Fast Fourier Transform (FFT) over $\mathbb{Q}$ can be efficiently implemented using complex roots of unity is that the cyclotomic extensions of the completion $\mathbb{R}$ of $\mathbb{Q}$ are at most quadratic, and that roots of unity in $\mathbb{C}$ can be evaluated quickly. In this paper, we investigate a $p$-adic analogue of this efficient FFT. A naive application of this idea--such as invoking well-known algorithms like the Cantor-Zassenhaus algorithm or Hensel's lemma for polynomials to compute roots of unity--would incur a cost quadratic in the degree of the input polynomial. This would eliminate the computational advantage of using FFT in the first place. We present a method for computing roots of unity with lower complexity than the FFT computation itself. This suggests the possibility of designing new FFT algorithms for rational numbers. As a simple application, we construct an $O(N^{1+o(1)})$-time FFT algorithm over $\mathbb{Q}_p$ for fixed $p$.