Home > Irreducible polynomial

an object cannot be expressed as a product of at least two non-trivial factors in a given ring. See also factorization.

For any field F, the ring of polynomials with coefficients in F is denoted by . A polynomial

irreducible over , if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from .

This definition depends on the field F. Some simple examples will be discussed below.

its Galois group, and its irreducible polynomials in depth. Interesting and non-trivial applications can be found in the study of finite fields.

It is helpful to compare irreducible polynomials to

corresponding negative numbers of equal modulus) are the irreducible integers. They exhibit many of the general properties of the concept 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of constants from F to the factors.

1 Simple examples

The following three polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

,

,

.

Over the field Q of rational numbers, the first polynomial is reducible, but the other two polynomials are irreducible.

Over the field R of real numbers, the two polynomials and are reducible, but is still irreducible.

Over the field C of complex numbers, all three polynomials are reducible.

In fact over C, every polynomial can be factored into linear factors

where is the leading coefficient of the polynomial and are the zeros of . Hence, all irreducible polynomials are of degree 1. This is the Fundamental theorem of algebraThe fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if : (where the coefficients a . a can be r.

Note: The existence of an essentially unique factorization

of into factors that do not belong to

over Q: there cannot be another factorization.

These examples demonstrate the relationship between the zeros of a polynomial (solutions of an algebraic equation) and the factorization of the polynomial into linear factors.

The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extensionIn abstract algebra, an extension of a field K is a field L which contains K as a subfield. For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers). of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers to real numbers and further to complex numbers.

For algebraic purposes, the extension from rational numbers to real numbers is often too 'radical': It introduces transcendental numberIn mathematics, a transcendental number is any irrational number that is not an algebraic number, i. it is not the solution of any polynomial equation of the form : where n ≥ 1 and the coefficients a are integers (or, equivalently, rationals), not alls (that are not the solutions of algebraic equations with rational coefficients). These numbers are not needed for the algebraic purpose of factorizing polynomials (but they are necessary for the use of real numbers in analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g). Thus, there is a purely algebraic process to extendIn abstract algebra, an extension of a field K is a field L which contains K as a subfield. For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers). a given field F with a given polynomial to a larger field where this polynomial can be reduced into linear factors. The study of such extensions is the starting point of Galois theory.