Irreducible (idea) by Noether

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.

Any polynomial of degree one is irreducible.
Over the complex numbers the irreducible polynomials are the polynomials of degree one. This is what the fundamental theorem of algebra says.
Over the real numbers the irreducible polynomials are the polynomials of degree one and the polynomials of degree two with no real root.
Over the rational numbers there are irreducible polynomials of any degree. For example, xⁿ-2 is irreducible, for all n>0

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.

Eisenstein's irreducibility criterion	prime	representations of Lie algebras	unique factorization domain
principal ideal domain	Fundamental theorem of algebra	All finite fields are isomorphic to GF(p^n)	Polynomial
Markov chain	simple	unit	Kronecker's Theorem
Proof of the uniqueness of splitting fields	irreducibility of cyclotomic polynomials	unique factorization of polynomials	Gauss's Lemma
rational functions in one variable	Gaussian integer	minimal polynomial	cyclotomic polynomial
ring theory	Separable	monism