分裂域是根式塔的充分必要条件

Sufficient and necessary conditions for splitting fields to be radical extensions

设f(x)是域F上次数大于0的多项式,E是f(x)在F上的分裂域。利用可解群和Galois理论,给出了E是F的根式塔的一些充分必要条件。证明了E是F的根式塔当且仅当(1)Gal(E/F)是可解群;(2)E包含[E:F]的全部素因子次本原单位根。

Let f（ x）∈F[ x]be a polynomial over a field F,and suppose that E is a splitting field of f（ x）over F. Using Galois theory and solvable group,we give some sufficient and necessary conditions for E to be a radical tower of F. It is proved that E is a radical tower of F if and only if,（1）Galois group Gal（E/F）is solvable;（2）E contains a primi-tive p-th root of unity for every prime divisor p of[ E：F].