Higher reciprocity laws and ternary linear recurrence sequences

In this paper, we describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree [Formula: see text]. As an application, we establish some divisibility properties of the associated ternary recurrence sequence by primes p, thus greatly extending recent work of Evink and Helminck [Tribonacci numbers and primes of the form [Formula: see text], Math. Slovaca 69(3) (2019) 521–532] and of Faisant [On the Padovan sequence (2019), https://arxiv.org/abs/1905.07702 ]. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo p.