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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.

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In the right hand column of this page you'll see a list of related questions that have been asked before; some of them look like they might be helpful. In particular, the question "About irreducible trinomials" has an answer pointing to a paper by Ljunggren. There is not an obvious route to an online copy, but the Math Review says that "The methods used are direct and elementary". – Neil Strickland Jun 19 '13 at 21:07
"direct and elementary" doesn't necessarily mean avoiding the use of elementary calculus, which I imagine is the only analysis Selmer uses (to get information about the real roots). So, I wouldn't say that this answers the question unless someone who has access to the paper can tell us more about what it uses. – John Pardon Jun 20 '13 at 0:37
I looked at the paper by Ljunggren ("On the irreducibility of certain trinomials and quadrinomials." Math. Scand. 8 1960 65–70.) He only uses basic algebra, as far as I can tell. – Mark Sapir Jun 20 '13 at 1:57
Here is the link to the paper: – Mark Sapir Jun 20 '13 at 2:04

For an algebraic approach to the irreducibility, see the sketch I wrote as an answer to

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