The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as
By the fundamental theorem of algebra, a univariate polynomial is absolutely irreducible if and only if its degree is one.
On the other hand, with several indeterminates, there are absolutely irreducible polynomials of any degree, such as
When the coefficient ring is a field or other unique factorization domain, an irreducible polynomial is also called a prime polynomial, because it generates a prime ideal.
If F is a field, a non-constant polynomial is irreducible over F if its coefficients belong to F and it cannot be factored into the product of two non-constant polynomials with coefficients in F. A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain R, is sometimes said to be irreducible (or irreducible over R) if it is an irreducible element of the polynomial ring, that is, it is not invertible, not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in R. This definition generalizes the definition given for the case of coefficients in a field, because, over a field, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero.
Another definition is frequently used, saying that a polynomial is irreducible over R if it is irreducible over the field of fractions of R (the field of rational numbers, if R is the integers).
The equivalence of the two definitions depends on R. The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials: Over the integers, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers); the last two are irreducible.
This fact is known as the fundamental theorem of algebra in the case of the complex numbers and, in general, as the condition of being algebraically closed.
There are irreducible multivariate polynomials of every degree over the complex numbers.
For example, the polynomial which defines a Fermat curve, is irreducible for every positive n. Over the field of reals, the degree of an irreducible univariate polynomial is either one or two.
Over a unique factorization domain the same theorem is true, but is more accurately formulated by using the notion of primitive polynomial.
A primitive polynomial over F is irreducible over F if and only if it is irreducible over the field of fractions of F. Every polynomial over F may be decomposed into the product of a non-zero constant and a finite number of non-constant irreducible primitive polynomials.
The non-zero constant may itself be decomposed into the product of a unit of F and a finite number of irreducible elements of F. Both factorizations are unique up to the order of the factors and the multiplication of the factors by a unit of F. This is this theorem which motivates that the definition of irreducible polynomial over a unique factorization domain often supposes that the polynomial is non-constant.
All algorithms which are presently implemented for factoring polynomials over the integers and over the rational numbers use this result (see Factorization of polynomials).
(that is, it is not the product of two non-constant polynomials with integer coefficients).
Eisenstein's criterion is a variant of this property where irreducibility over
The converse, however, is not true: there are polynomials of arbitrarily large degree that are irreducible over the integers and reducible over every finite field.
The relationship between irreducibility over the integers and irreducibility modulo p is deeper than the previous result: to date, all implemented algorithms for factorization and irreducibility over the integers and over the rational numbers use the factorization over finite fields as a subroutine.
The number of degree n irreducible monic polynomials over a field
For q = 2, such polynomials are commonly used to generate pseudorandom binary sequences.
In some sense, almost all polynomials with coefficients zero or one are irreducible over the integers.
More precisely, if a version of the Riemann hypothesis for Dedekind zeta functions is assumed, the probability of being irreducible over the integers for a polynomial with random coefficients in {0, 1} tends to one when the degree increases.
Even the irreducibility of a polynomial may not always be proved by a computation: there are fields over which no algorithm can exist for deciding the irreducibility of arbitrary polynomials.
[8] Algorithms for factoring polynomials and deciding irreducibility are known and implemented in computer algebra systems for polynomials over the integers, the rational numbers, finite fields and finitely generated field extension of these fields.
The notions of irreducible polynomial and of algebraic field extension are strongly related, in the following way.
Let x be an element of an extension L of a field K. This element is said to be algebraic if it is a root of a nonzero polynomial with coefficients in K. Among the polynomials of which x is a root, there is exactly one which is monic and of minimal degree, called the minimal polynomial of x.
by the ideal generated by P. Then L is a field if and only if P is irreducible over K. In this case, if x is the image of X in L, the minimal polynomial of x is the quotient of P by its leading coefficient.
If a polynomial P has an irreducible factor Q over K, which has a degree greater than one, one may apply to Q the preceding construction of an algebraic extension, to get an extension in which P has at least one more root than in K. Iterating this construction, one gets eventually a field over which P factors into linear factors.
One can show that every prime element is irreducible;[9] the converse is not true in general but holds in unique factorization domains.