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In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
In mathematics, an irreducible polynomial is, roughly speaking, a 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 ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
Consider the polynomial Q(x) = 3x 4 + 15x 2 + 10.In order for Eisenstein's criterion to apply for a prime number p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10.
Because of this problem of undecidability in the formal language of computation, Wolfram terms this inability to "shortcut" a system (or "program"), or otherwise describe its behavior in a simple way, "computational irreducibility." The idea demonstrates that there are occurrences where theory's predictions are effectively not possible.
Gauss's lemma (irreducibility) — A non-constant polynomial in Z[X] is irreducible in Z[X] if and only if it is both irreducible in Q[X] and primitive in Z[X]. The proof is given below for the more general case.
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element in a commutative ring is called prime if, whenever for some and in , then or . ...
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place.
Absolute irreducibility more generally holds over any field not of characteristic two. In characteristic two, the equation is equivalent to (x + y −1) 2 = 0. Hence it defines the double line x + y =1, which is a non-reduced scheme. The algebraic variety given by the equation + = is not absolutely irreducible.