mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property...

Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not...

product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite...

computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically...

K[X]/(p)} is a field if and only if p is an irreducible polynomial. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p...

polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials)...

In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor...

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...

GF(pm). Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. A primitive polynomial must have a non-zero constant...

characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree...

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...

of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors...

There is at least one irreducible polynomial for which x is a primitive element. In other words, for a primitive polynomial, the powers of x generate...

p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (that is not factorable into polynomials of positive...

primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has...

F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditions for the irreducible polynomial f to be separable:...

lemma (polynomial) Irreducible polynomial Eisenstein's criterion Primitive polynomial Fundamental theorem of algebra Hurwitz polynomial Polynomial transformation...

Nn be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from...

P(x) is an additive polynomial. Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients...

theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number...

in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then...

^{2}=2\alpha ^{5}+\alpha ^{2}+2\alpha .} Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is given...

normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors in L. This...

In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define...

one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are...

− x ( x − 1 ) ( x − 2 ) {\displaystyle y^{2}-x(x-1)(x-2)} is an irreducible polynomial. The ring Z [ x ] / ( x 2 − n ) ≅ Z [ n ] {\displaystyle \mathbb...

{-5}}\right)=9,} but 3 does not divide either of the two factors. Irreducible polynomial Consider p {\displaystyle p} a prime element of R {\displaystyle...

stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials. Vieta's formulas...

numerator and the denominator are coprime polynomials. Every rational number can be represented as an irreducible fraction with positive denominator in exactly...

Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} —that is...