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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.
A 3-manifold is P²-irreducible if it is irreducible and contains no 2-sided (real projective plane). An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.
The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one. The assertion "the polynomials of degree one are irreducible" is trivially true for any field. If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore ...
In algebra, a field k is perfect if any one of the following equivalent conditions holds: Every irreducible polynomial over k has no multiple roots in any field extension F/k. Every irreducible polynomial over k has non-zero formal derivative. Every irreducible polynomial over k is separable. Every finite extension of k is separable.
If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x 2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root.
Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n).
A univariate polynomial of degree greater than or equal to 2 is never absolutely irreducible, due to the fundamental theorem of algebra.; The irreducible two-dimensional representation of the symmetric group S 3 of order 6, originally defined over the field of rational numbers, is absolutely irreducible.