Irreducible polynomial

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 $\mathbb{Z}[x]$.