On Eisenstein–Dumas and Generalized Schönemann Polynomials
On Eisenstein–Dumas and Generalized Schönemann Polynomials
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关于爱森斯坦-杜马斯和广义舍内曼多项式
DOI：
10.1080/00927870903164669
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发表时间：
2010
影响因子：
0.7
通讯作者：
S. K. Khanduja
中科院分区：
文献类型：
--
作者：
Anuj Bishnoi;S. K. Khanduja
设v是具有值群ℤ的域K的赋值。已知多项式x n+a n−1 x n−1+…满足v(A0)互素到n的+a0在K上是不可约的，这样的多项式称为关于v的Eisenstein-Dumas多项式.本文给出了属于K[x]的给定多项式g(x+a)的某个平移g(x+a)是关于v的Eisenstein-Dumas多项式的充要条件.实际上，对于更广泛的多项式，即Z，讨论了一个类似的问题.任意秩值域上系数的广义Schönemann多项式。
Let v be a valuation of a field K having value group ℤ. It is known that a polynomial x n + a n−1 x n−1 + … +a 0 satisfying with v(a 0) coprime to n, is irreducible over K. Such a polynomial is referred to as an Eisenstein–Dumas polynomial with respect to v. In this article, we give necessary and sufficient conditions so that some translate g(x + a) of a given polynomial g(x) belonging to K[x] is an Eisenstein–Dumas polynomial with respect to v. In fact, an analogous problem is dealt with for a wider class of polynomials, viz. Generalized Schönemann polynomials with coefficients over valued fields of arbitrary rank.