Has the problem of factoring (over the rationals) the general trinomial $ax^n+bx^k+c$ with $a,b,c\in\mathbb{Z}$, $n,k\in\mathbb{N}, n>k>1$ been solved? By solved I mean a decision procedure classification theorem which will either factor give the factors or give a certificate that it is irreducible.
One could ask the same question, but about finding expressions for the roots of a trinomial. I am aware that long ago this problem was worked on, but I have been unable to find a modern treatment of this problem.
The question which actually motivates this one is: given a trinomial which factors, consider the number of terms in each of the factors. Further, assume that the trinomial is not cyclotomic. How many terms can the factors have? [The cyclotomic case is conjectured to be special].
[Edit: clarified that I am interested in this special case, not the general case, where the algorithms of M. van Hoeij are the current best, much superior to Cantor-Zassenhaus].
