On the embedding problem for nonsolvable Galois groups of algebraic number fields: Reduction theorems

AbstractThe embedding problem, which is the problem of extending a given Galois extension K ⊃ k to a Galois extension L ⊃ K ⊃ k so that G(Lk) is a prescribed group extension of G(Kk), is investigated in the case k is a number field and G(LK) is nonsolvable, with respect to the question of reduction methods. Two general (arbitrary k and G(LK) reduction theorems are proved, one reducing the general problem to the cases of G(LK) nilpotent, and split group extensions, resp, and the second reducing the problem in the case G(LK) having trivial center to the case G(Lk)∼⊂ aut G(LK). The notion of localizability of an embedding problem is formulated and investigated for certain classical groups