  We describe the set of prime numbers splitting completely in the non-abelian
splitting field of certain monic irreducible polynomials of degree three. 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 and of Faisant. We also prove some new results on the number of
solutions of the characteristic equation of the recurrence sequence modulo $p,$
extending and simplifying earlier work of Zhi-Hong Sun (2003).
