Contents Definition edit 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.
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.
; Van Oorschot, Paul.
Then L is a field if and only if P is irreducible over.When a polynomial is reducible into factors, these factors may be explicit algebraic expressions or implicit expressions.Displaystyle mathbb C mathbb R X X21).A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a greatest common divisor of its coefficients.A polynomial that is not irreducible is sometimes said to be reducible.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.Conversely, if P ( X ) K X displaystyle P(X)in KX is a univariate polynomial over a field K, let L K X / P ( X ) displaystyle LKX/P(X) be the"ent ring reduction holiday on ice of the polynomial ring K X displaystyle KX by the ideal.John Wiley Sons, Inc.On the other hand, with several indeterminates, there are coffret cadeau amor amor absolutely irreducible polynomials of any degree, such as x 2 y n 1, displaystyle x2yn-1, for any positive integer.6 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 converse, however, is not true: there are polynomials of arbitrarily large degree that are irreducible over the integers and reducible over every finite field.
Let x be an element of an extension L of a field.
Nature of a factor edit The absence of an explicit algebraic expression for a factor does not by itself imply that a polynomial is irreducible.
Both factorizations are unique up to the order of the factors and the multiplication of the factors by a unit.More precisely, the irreducible polynomials are the polynomials of degree one and the quadratic polynomials a x 2 b x c displaystyle ax2bxc that have a negative discriminant b 2 4.The polynomial ring F x over a field F (or any unique-factorization domain) is again a unique factorization domain.A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible.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.The non-zero constant may itself be decomposed into the product of a unit of F and a finite number of irreducible elements.