Efficient computation of primitive M-th roots of unity in low-degree ℤ_p-extensions

Determine algorithmic methods and complexity bounds for computing a primitive M-th root of unity ζ_M within low-degree algebraic extensions of the p-adic integers ℤ_p, for a prime p and integer M coprime to p, so that Cooley–Tukey FFT over the p-adic field ℚ_p can be executed when ζ_M does not lie in ℤ_p itself.

Background

The discussion contrasts classical number-theoretic transforms over finite fields with the p-adic setting. When M divides p−1, a primitive M-th root of unity lies in ℤ_p and can be found efficiently via simple-root Hensel lifting, enabling FFT. However, to leverage small-degree cyclotomic extensions for broader M, one must compute ζ_M inside low-degree algebraic extensions of ℤ_p.

The paper points out that, unlike the straightforward ℤ_p case, the complexity and practical methods for constructing ζ_M in low-degree extensions of ℤ_p are not established. This uncertainty motivates developing algorithms to efficiently construct such roots of unity for p-adic FFTs.