On the imbedding problem with noncommutative kernel of order p4. III

Abstract

In the case of number fields the embedding problem of a p-extension with non-Abelian kernel of order p4 is studied. The two kernels of order 34 with generators α, γ and relations α9 = 1, [α,α]3=1,[α,αγγ]==1,[αγγ]=α3,γ3=1 or γ3=α3 and the kernel of order 24 with generators α, β, γ and relations α4=1 β2,[αγ]=1, [α,γ]=1,[βγ]=α2 are considered. For kernels of odd order the embedding problem is always solvable. For the kernel of order 16 the solvability conditions are reduced to those for the associated problems at the Archimedean points, and to the compatibility condition. Bibliography: 9 titles.