Irreducible polynomial

Let $R$ be an integral domain . A polynomial $f\in R[x]$ is irreducible if $f$ is nonzero, not a unit , has positive degree, and whenever $f=gh$ with $g,h\in R[x]$, then either $g$ or $h$ is a unit.

Over a field, irreducible polynomials generate maximal ideals in $R[x]$ and define finite field extensions. Practical tests include Eisenstein’s criterion , and primitive hypotheses are often used to compare factorization in $R[x]$ and in $\mathrm{Frac}(R)[x]$.

Examples:

• In $\mathbb{R}[x]$, the polynomial $x^2+1$ is irreducible.
• In $\mathbb{Q}[x]$, $x^3-2$ is irreducible (Eisenstein at $2$).
• In any $R[x]$, $x^2-1=(x-1)(x+1)$ is reducible.