Irreducibility of All One Polynomial
Theorem
The all one polynomial of degree \(n\) is irreducible over \(\mathbb{Q}\) if \(n + 1\) is prime.
Proof
This follows from the irreducibility of the cyclotomic polyonmials, and the fact that \(\Phi_p(X)\) is the all one polynomial of degree \(p - 1\) for prime \(p\).