Extending Some Irreducibility Results of Finch and Jones

摘要

For any list T = [s, t, n] of positive integers, define the polynomial F_T(x) = x~nγ_s(x)+γ_t(x)∈Z[x], where γ_s(x) =∑_(i=0)~s x~i Finch and Jones gave sufficient conditions for the reducibility of Fr(x) over Q, based solely on the elements of T, and they conjectured that these conditions are also necessary. In this article, we establish this conjecture in a more generalized context. In addition, we extend two other results from the original paper of Finch and Jones.