分裂域和重复根式扩张

Splitting Fields and Repeated Radical Extensions

设E为系数在F上的多荐式f(x)的分裂域,若f(x)在F上根式可解,则E必含在F的一个重复根式扩张中,而E不一定是F的重复根式扩张。本文继续探讨了这一问题,当CharF n=deg(f(x))时证明:(1)若Galiois群Gal(E/F)可解,E包含pi次本原单位根,则E是F的重复根式扩张,这里pi是deg(f(x))的全部素因子;(2)若E是F的重复根式塔,则E包含pi次本原单位根;并讨论了n=2spt11pt22…ptkk,pi为Fermat素数的情形。

Let f(x) be non_constant polynomial over a field F with E its splitting field. If f(x)=0 can be solved radically over F,E is contained in a repeated radical extension,while it not necessarily a repeated radical extension itself. When CharFn=deg(f(x)),this paper continues the discussion and proves that:(1)If the Galois group Gal(E/F)is solvable and E contains the p_i_th primitive roots of unity,then E is a repeated radical extension,where,p_i is the prime factor of n;(2)If E is a repeated radical tower,then E contains the p_i_th primitive roots of unity;Cases like n=2~sp^(t_1)_1p^(t_2)_2...p^(t_k)_k,p_i is a Fermat prime,are also discussed.