What is content of a polynomial? — Mathematics

Definition 2.22 University Mathematics — Year 3 · Chapter 2 — Rings and Arithmetic

The content $c(P)$ of a nonzero $P \in A[X]$ is a gcd of its coefficients (defined up to a unit); $P$ is primitive if $c(P) \in A^\times$ . Every $P \in A[X]$ writes $P = c(P)\,P_1$ with $P_1$ primitive, and every $P \in K[X]\setminus\{0\}$ writes $P = \lambda P_1$ with $\lambda \in K^\times$ and $P_1 \in A[X]$ primitive (clear denominators, then factor out the content).

Examples

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

\Phi_p(X + 1) = \frac{(X+1)^p - 1}{X} = X^{p-1} + \binom{p}{1}X^{p-2} + \dots + \binom{p}{p-1},

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.

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