Irreducible Polynomial

Irreducible Polynomial

In mathematics, a polynomial is said to be irreducible if it cannot be factored into the product of two or more non-trivial polynomials whose coefficients are of a specified type. Thus in the common context of polynomials with rational coefficients, a polynomial is irreducible if it cannot be expressed as the product of two or more such polynomials, each of them having a lower degree than the original one. For example, while is reducible over the rationals, is not.

For any field F, a polynomial with coefficients in F is said irreducible over F if it is non-constant and cannot be factored into the product of two or more non-constant polynomials with coefficients in F. The property of irreducibility depends on the field F; a polynomial may be irreducible over some fields but reducible over others. Some simple examples are discussed below.

It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:

Every polynomial with coefficients in a field F can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F. This property of unique factorization is commonly expressed by saying that the polynomial rings over a field are unique factorization domains. However the existence of such a factorization does not mean that, given a polynomial, the factorization may always be computed: there are fields such that it can not exist any algorithm to factorize polynomials over these fields.. There exist factorization algorithms for the polynomials with coefficients in the rational numbers, in a finite field or a finitely generated field extension of these fields. They are described in the article Polynomial factorization.

If an univariate polynomial p has a root (in some field extension) which is also a root of an irreducible polynomial q, then p is a multiple of q, and thus all roots of q are roots of p; this is Abel's irreducibility theorem. This implies that the roots of an irreducible polynomial may not be distinguished through algebraic relations. This result is one of the starting points of Galois theory, which has been introduced by Évariste Galois to study the relationship between the roots of a polynomial.