Prove that $P_n(X) = X^n - X^{n-1} - X^{n-2} - ... - X - 1$ is irreducible over $\mathbb{Z}$ for all $n$.
I was able to prove the result for $n=2^k-1$ by applying Eisenstein's criterion to $P_n(X+1)$. But for other values of $n$, I'm stuck. Has anyone an idea on this ?
Use the same idea used in the proof of Perron's irreducibility criterion:
Prove that $P_n(x)$ has one and only one root $a$ with $|a|>1$ and none with $|a|=1$ (this is the difficult part. To prove this use that $(x-1)P_n(x)=x^{n+1}-2x^n+1$).
If $P_n(x)$ was reducible with $P_n(x)=f(x)g(x)$ ($1\leq\deg(f)<n$) then one of the polynomials $f,g$ has all of its roots inside the unit circle and constant term $\pm1 \ \Rightarrow\Leftarrow$.