What is content of a polynomial?

Example 2.26

$X^n - p$ is irreducible over $\Q$ for every prime $p$ and $n \geq 1$ (Eisenstein at $p$): there are irreducible polynomials of every degree over $\Q$ — in stark contrast with $\C$ (degree $1$, d’Alembert–Gauss, proved in Chapter 16) and $\R$ (degrees $1, 2$). The trick of shifting enlarges Eisenstein’s reach: the $p$-th cyclotomic polynomial $\Phi_p = X^{p-1} + \dots + X + 1 = \frac{X^p - 1}{X - 1}$ has

Eisenstein at $p$ ($p \mid \binom pk$ for $0 < k < p$, and $\binom{p}{p-1} = p \not\equiv 0 \bmod p^2$): $\Phi_p(X+1)$, hence $\Phi_p$, is irreducible over $\Q$. This is the algebraic heart of the $17$-gon story told in Chapter 4.