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- Eisenstein's_criterion abstract "In mathematics, 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 be unfactorable into the product of non constant polynomials with rational coefficients. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.Suppose we have the following polynomial with integer coefficients. If there exists a prime number p such that the following three conditions all apply:p divides each ai for i ≠ n,p does not divide an, andp2 does not divide a0,then Q is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case Q as integer polynomial will have some prime number, necessarily distinct from p, as an irreducible factor). The latter possibility can be avoided by first making Q primitive, by dividing it by the greatest common divisor of its coefficients (the content of Q). This division does not change whether Q is reducible or not over the rational numbers (see Primitive part–content factorization for details), and will not invalidate the hypotheses of the criterion for p (on the contrary it could make the criterion hold for some prime, even if it did not before the division). This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort. In some cases the criterion does not apply directly (for any prime number), but it does apply after transformation of the polynomial, in such a way that irreducibility of the original polynomial can be concluded.".
- Eisenstein's_criterion wikiPageID "385989".
- Eisenstein's_criterion wikiPageRevisionID "596683324".
- Eisenstein's_criterion hasPhotoCollection Eisenstein's_criterion.
- Eisenstein's_criterion id "a/a011480".
- Eisenstein's_criterion title "Algebraic equation".
- Eisenstein's_criterion subject Category:Articles_containing_proofs.
- Eisenstein's_criterion subject Category:Field_theory.
- Eisenstein's_criterion subject Category:Polynomials.
- Eisenstein's_criterion type Abstraction100002137.
- Eisenstein's_criterion type Function113783816.
- Eisenstein's_criterion type MathematicalRelation113783581.
- Eisenstein's_criterion type Polynomial105861855.
- Eisenstein's_criterion type Polynomials.
- Eisenstein's_criterion type Relation100031921.
- Eisenstein's_criterion comment "In mathematics, 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 be unfactorable into the product of non constant polynomials with rational coefficients. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.Suppose we have the following polynomial with integer coefficients.".
- Eisenstein's_criterion label "Criterio de Eisenstein".
- Eisenstein's_criterion label "Criterio di Eisenstein".
- Eisenstein's_criterion label "Criterium van Eisenstein".
- Eisenstein's_criterion label "Critère d'Eisenstein".
- Eisenstein's_criterion label "Critério de Eisenstein".
- Eisenstein's_criterion label "Eisenstein's criterion".
- Eisenstein's_criterion label "Eisensteinkriterium".
- Eisenstein's_criterion label "Kryterium Eisensteina".
- Eisenstein's_criterion label "Критерий Эйзенштейна".
- Eisenstein's_criterion label "艾森斯坦判別法".
- Eisenstein's_criterion sameAs Eisensteinkriterium.
- Eisenstein's_criterion sameAs Criterio_de_Eisenstein.
- Eisenstein's_criterion sameAs Critère_d'Eisenstein.
- Eisenstein's_criterion sameAs Criterio_di_Eisenstein.
- Eisenstein's_criterion sameAs 아이젠슈타인_기준.
- Eisenstein's_criterion sameAs Criterium_van_Eisenstein.
- Eisenstein's_criterion sameAs Kryterium_Eisensteina.
- Eisenstein's_criterion sameAs Critério_de_Eisenstein.
- Eisenstein's_criterion sameAs m.0225t1.
- Eisenstein's_criterion sameAs Q1057416.
- Eisenstein's_criterion sameAs Q1057416.
- Eisenstein's_criterion sameAs Eisenstein's_criterion.
- Eisenstein's_criterion wasDerivedFrom Eisenstein's_criterion?oldid=596683324.
- Eisenstein's_criterion isPrimaryTopicOf Eisenstein's_criterion.