Often called prime polynomials. Polynomials that have no polynomial divisors.
An irreducible or prime polynomial is a polynomial that has no polynomial factors. The parallel with prime integers is appropriate. In the same way that a prime number cannot be divided by any other integer other than $\pm1$, an irreducible polynomial cannot be divided by any polynomial other than a constant non-zero polynomial.
Though it is usually clear from context, care should be taken in observing the domain over which a polynomial is irreducible. For instance, $x^2+4x+13$ is irreducible over $\mathbb{R}$ but not $\mathbb{C}$; likewise, $x^3 + 2 x^2 + x + 3$ is irreducible over $\mathbb{F}_{5}$ but factors as $(x-1)(x-2)^2$ over $\mathbb{F}_7$.
