|
In mathematics, Eisenstein This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases to prove irreducibility with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it.〔〔.〕 ==Criterion== Suppose we have the following polynomial with integer coefficients. : If there exists a prime number such that the following three conditions all apply: * divides each for , * does ''not'' divide , and * does ''not'' divide , then 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 as integer polynomial will have some prime number, necessarily distinct from , as an irreducible factor). The latter possibility can be avoided by first making primitive, by dividing it by the greatest common divisor of its coefficients (the content of ). This division does not change whether 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 (on the contrary it could make the criterion hold for some prime, even if it did not before the division). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Eisenstein's criterion」の詳細全文を読む スポンサード リンク
|