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Dear Professors!

Thank you for the most interesting paper (arXiv:2002.09472v2).

I would like to submit the following comment to you, counting on your benevolence in case I am mistaken here, as I have long exited math and am talking out of dim memories.

In section 8.2 'Modular roots', it seems to me that both Lemma 8.5 and Theorem 8.7 may be viewed as applications of Cebotarev's density theorem (which is a fixture of algebraic number theory textbooks).

In the case of Lemma 8.5, apply Cebotarev's theorem to the Galois extension of Q generated by S (that is, Q(S') where S' consists of the elements of S together with all their images under automorphisms of the algebraic closure of Q), and the conjugacy class of the identity in Gal(Q(S')/Q). Then, if I'm not mistaken, Cebotarev's theorem says that the set of primes that split completely in Q(S') has positive density, namely 1/[Q(S') : Q], and any such completely-split prime satisfies the requirement of Lemma 8.5, namely, the desired map from Z[S] to F_p is obtained as the composition of the following maps:
Z[S] -> Z[S'] -> Z[S']/pZ[S'] = Z[S']/(p_1 ... p_n) -> Z[S']/p_1 = F_p

If I understand correctly, Theorem 8.7 was only used to prove Lemma 8.5, so the above might remove the need for it (and make your paper more self-contained if you allow using Cebotarev's theorem). However, if for whatever reason you still need Theorem 8.7, then I believe that it too is a direct application of Cebotarev's theorem. Actually the wikipedia page on it mentioned a similar statement as having been discovered earlier by Frobenius, as a predecessor to Cebotarev's theorem, in this paragraph.

Cheers,
Benoit Jacob