Consider the following family of polynomials over $\mathbb{Q}$:


$$f\_n = x^n - x^{n-1} - \dots - 1$$


Notice that these polynomials satisfy the recurrence


$$ f\_{n+1} = x f\_n - 1 $$


I would like to show that every polynomial in this family is irreducible over $\mathbb{Q}$. My strategy for showing this is to show that for each of these polynomials, all but one of the roots are in the unit circle (this would imply irreducibility since if $f\_n(x) = g(x) h(x)$, then since $f\_n$ has only one root of size greater than one, only one of $g, h$ can have non-zero constant term. This implies that 0 is a root of either $g$ or $h$, but 0 is not a root of $f\_n$ for any $n$).


I am attempting to proceed by induction using Rouche's theorem. The idea is that if $f\_n$ has $n-1$ roots in the unit circle, then $x f\_n$ has $n$ roots in the unit circle. Therefore, if it's true that


$$ |f\_{n+1} - x f\_n | = 1 < |f\_{n+1}| + |x f\_{n}| $$


for all $x$ on the unit circle, then $f\_{n+1}$ has $n$ roots inside the unit circle. However, I am not able to prove that we always have strict inequality above.


I suspect that this family $f\_n$ has been studied before, but I was not able to find it in the literature. Any help would be appreciated.



Irreducibility of family of polynomials
This family of polynomials is studied in [The Galois group of $x^n - x^{n-1} - \cdots - x - 1$](http://www.sciencedirect.com/science/article/pii/S0022404903002457) by Paulo Martin. It is shown that each $f\_n$ is irreducible in Corollary 2.2.


