Proving the Irreducibility of Multinacci Polynomial g(x) for Natural Numbers m

[Diophantus]

Can anyone please give me a hint on how I can prove that

$$g(x) = x^m - x^{m-1} - x^{m-2} - ... - x - 1$$

is irreudicble over the rationals for all natural numbers m?

Regards

 

[Oxymoron]

I would use an evaluation homomorphism, to change the variable, and then use the Eisenstein criterion for a prime p and show it is irreducible. I haven't actually done it (so this might be wrong), but this is what I would try first.

The evaluation homomorphism I would use would be of the form

$$\phi_{x+1}\,:\,\mathbb{Q}[x] \rightarrow \mathbb{Q}[x] \quad\quad \phi_{x+1}(g(x)) = g(x+1)$$

Hopefully you'll get a nice binomial expression to which you can check if p divides (by Eisenstein's Criteria).

 

[Diophantus]

Alas, the Eisenstein criteria will not work here for every m.

Indeed $$\phi_{x+1}(x^4 - x^3 - x^2 - x - 1) = x^4 + 3x^3 + 2x^2 - 2x - 3$$

and then (2,3) = 1 implies that no suitable prime can be found.

Any other ideas?