I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain trinomials", and I got his idea, but... In his proof he uses analysis methods. I wonder, if the pure algebraic proof exists? I mean, without making graphics and curves, but studying polynomial as the element of Q[x], not as a function.
For an algebraic approach to the irreducibility, see the sketch I wrote as an answer to http://math.stackexchange.com/questions/393646/irreducibility-of-xn-x-1-over-mathbb-q.