Irreducible polynomial

Definition

Let $R$ be an integral domain. A polynomial $f \in R[x]$ is irreducible over $R$ if it is an irreducible element of the integral domain $R[x]$. That is, $f$ is nonzero and not a unit, and if $f = g h$ with $g, h \in R[x]$, then $g$ or $h$ is a unit of $R[x]$.

Remarks

Over a field $K$, the units of $K[x]$ are the nonzero constants. So $f$ is irreducible if and only if $\deg f \ge 1$ and $f$ is not a product of two polynomials of smaller degree. The polynomial $x^2 - 2$ is irreducible over $\mathbb{Q}$ but not over $\mathbb{R}$, where $x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})$. The polynomial $x^2 + 1$ is irreducible over $\mathbb{R}$ but not over $\mathbb{C}$. Over $\mathbb{Z}$, the polynomial $2x + 2 = 2(x + 1)$ is not irreducible, because $2$ is not a unit of $\mathbb{Z}[x]$.

Explanations

L1 human

Irreducible polynomials are the prime numbers of polynomial rings. Over a field, they are the polynomials that you cannot factor into polynomials of smaller degree. Irreducibility depends on the field: $x^2 + 1$ factors over $\mathbb{C}$ but not over $\mathbb{R}$.