Title: On extensions allowing a Galois theory and their Galois groups
Abstract: Let E / F be a field extension and let G =Aut( E / F ). E / F is said to allow a Galois theory if there is a subgroup G ′ of G such that the map H → E H (the fixed field of H ) is a bijection between the set of all subgroups H of G ′ and the set of all intermediate fields of E / F . Such extensions E / F and their Galois groups were characterized in several ways in [2]. In this paper, first a conjecture on G being a prime sparse product of finite groups is settled in the negative. Secondly it is show however that G = G 1 · G 2 where G 1 , G 2 are such products. Further there is an intermediate field N normal over F such that each element of N is solvable by radicals over F and E is finite over N . Several properties of G and E / F are also obtained.