Why specifically github? Anyone sharing code must have a github account and use git?

Nice! I find it remarkable that, though the question is intended for signal processing purposes, Gronwall was tightening a bound of Landau that was used to study a number theory problem, namely about what are now called Fekete polynomials, that come up while studying class numbers of quadratic fields. (See reference in loc. cit.)

I find this question to be interesting, though the subject matter is not my field of expertise. But I do have two things to report on, both from Montgomery's paper An Exponential Polynomial Formed with the Lagrange Symbol: 1. the Fekete polynomials give a lower bound $\gamma>2\log\log n/\pi$, so there's no uniform constant $\gamma$. 2. as Montgomery writes, Bernstein's inequality give a bound in terms of "a higher sampling rate", i.e. the polynomial evaluated at roots of unity of higher order. He doesn't prove the bound in the paper, but I think it works for all real polynomials.

WLOG scale so that the norm is 1, and then we have the inequalities $1/\sqrt{N}\le m_d\le m_c \le 1$. So at the least you can take $\gamma=\sqrt{N}$. Since at each of the inequalities we must have $m_c=m_d$, I'm guessing there's a much better bound...

A naive application of the abc conjecture gives $v_p((p-1)!+1)=o(p)$.

For each irreducible factor $g|f$ you can compute and factor the discriminant, giving you constraints on the possible $n$ for which $g$ factorizes over $\mathbb{Q}(\zeta_n)$.

@Julian: factorials can be computed in square root time using fast polynomial evaluation, so the binomial coefficient identity gives a $O(\sqrt{p})$ algorithm :) My guess is that this can be achieved for general rational functions.

Note that the theorem in Kaplansky's seems to only need torsion, so the above suffices for the intended application.

Would be cool if the answer satisfies an analogue of the Fontaine-Winterberger galois group isomorphism :P