Irreducible polynomial

In mathematics, an irreducible polynomial (or prime polynomial) is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as ( x − 2 ) ( x + 2 ) {displaystyle (x-{sqrt {2}})(x+{sqrt {2}})} if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals. In mathematics, an irreducible polynomial (or prime polynomial) is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as ( x − 2 ) ( x + 2 ) {displaystyle (x-{sqrt {2}})(x+{sqrt {2}})} if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals. A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible. 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 x 2 + y n − 1 , {displaystyle x^{2}+y^{n}-1,} for any positive integer n. A polynomial that is not irreducible is sometimes said to be reducible. However, this term must be used with care, as it may refer to other notions of reduction. Irreducible polynomials appear naturally in the study of polynomial factorization and algebraic field extensions. 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. 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). This second definition is not used in this article. The absence of an explicit algebraic expression for a factor does not by itself imply that a polynomial is irreducible. When a polynomial is reducible into factors, these factors may be explicit algebraic expressions or implicit expressions. For example, x 2 + 2 {displaystyle x^{2}+2} can be factored explicitly over the complex numbers as ( x − 2 i ) ( x + 2 i ) ; {displaystyle (x-{sqrt {2}}i)(x+{sqrt {2}}i);} however, the Abel–Ruffini theorem states that there are polynomials of any degree greater than 4 for which complex factors exist that have no explicit algebraic expression. Such a factor can be written simply as, say, ( x − x 1 ) , {displaystyle (x-x_{1}),} where x 1 {displaystyle x_{1}} is defined implicitly as a particular solution of the equation that sets the polynomial equal to 0. Further, factors of either type can also be expressed as numerical approximations obtainable by root-finding algorithms, for example as ( x − 1.2837... ) . {displaystyle (x-1.2837...).}