inverse Galois problem for and . Here is a start. If is an irreducible polynomial over with prime degree such that it has exactly two non-real zeros then form the splitting field . We argue that the group . First, note that is a subgroup of the permutation group of the zeros. Second, let be complex-conjugation, then and it leaves all real zeros intact, therefore, can be viewed as a transposition. Third, let be an automorphism which premutes the zeros in a cyclic fashion then can be regarded as a cycle of length . Thus, we have that has a cycle of length and a transpotion, thus in fact, .
(The link provides how to get as an inverse Galois problem).
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