**Def** A

polynomial over a

field *k* (like the

rational numbers,

real numbers or

complex numbers) is called

irreducible if it is not a

constant and cannot be

factored as a

product of polynomials (over

*k*) of smaller degree.

More generally, an element of a commutative integral domain *R* is called
irreducible if it is a non-unit and it cannot be written
as a product of two non-units in *R*.

For example, in **Z**, the ring of integers, the irreducible elements
are the prime numbers and their negatives.

If *a* is irreducible then any associate of *a* is irreducible.
See also prime.