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Sufficient and necessary conditions for splitting fields to #br# be radical extensions#br#

ZHANG Wen-hua1, JIANG Xiao-long2   

  1. 1. Department of Mathematics, Jining University, Qufu 273155, Shandong, China;
    2. Department of Mathematics, Sun Yatsen University, Guangzhou 510275, Guangdong, China
  • Received:2013-12-19 Online:2014-05-20 Published:2014-06-04

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Abstract: 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 primitive p-th root of unity for every prime divisor p of ［E:F］.

Key words: solvable group, splitting field, radical tower, radical extension

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