Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.