再论多项式的Hensel提升

Discussion on the Hensel lift of a polynomial

R是有限链环,M是其极大理想,K=R/M;则建立了K[x]中一类多项式在R[x]中的Hensel提升;证明了多项式的Hensel提升不依赖于n的选择,证明了K[x]中任一首一多项式f(x)在R[x]中具有Hensel提升的充要条件是f(0)≠0且f(x)在其分裂域中无重根。

Let R be a finite chain ring with the unique maximal ideal M. K denotes the residue field R/ M. The Hensel lifts in R[x] of certain polynomials in K[x] are given. It is proved that the Hensel lift is independent of the particular choice of n and that, for an arbitrary monic polynomial f（x） in K[x] , the Hensel lift in R[x] of f（x） exists if and only if f（0） ≠0 and f（x） has no multiple roots in its splitting field.