Ishkhanov, V. V.; Lur’e, B. B. On the embedding problem with noncommutative kernel of order \(p^4\). VI. (English. Russian original) Zbl 0974.12007 J. Math. Sci., New York 89, No. 2, 1127-1132 (1998); translation from Zap. Nauchn. Semin. POMI 227, 74-82 (1995). In Parts I–V the authors have studied the embedding problem for algebraic number fields with nonabelian kernel of order \(p^4\) (see Zbl 0871.12004 for Part V). However three possible kernels have been omitted so far: two of order 81 and one of order 16. It is shown that in the case of the first two kernels the problem is solvable both in the local and the global case. This is proved in case of the ground field contains the primitive cubic root of unity. When the kernel has 16 elements, necessary and sufficient conditions for the solvability of the embedding problem are given. Cited in 1 Review MSC: 12F10 Separable extensions, Galois theory 11R32 Galois theory 12F12 Inverse Galois theory Keywords:algebraic number fields; nonabelian kernel; embedding problem Citations:Zbl 0871.12004 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] V. V. Ishkhanov, ”On the embedding problem with non-Abelian kernel of orderp 4,”Trudy Math. Inst. Akad. Nauk SSSR,183, 116–121 (1990). [2] V. V. Ishkhanov and B. B. Lur’e, ”On the embedding problem with non-Abelian kernel of orderp 4,”Zap. Nauchn. Semin. LOMI,175, 46–62 (1989). [3] V. V. Ishkhanov and B. B. Lur’e, ”The embedding problem for number fields with noncommutative kernel of orderp 4,”Algebra Analiz,2, No. 6, 161–167 (1990). [4] V. V. Ishkhanov and B. B. Lur’e, ”On the embedding problem with non-Abelian kernel of orderp 4. II,”Zap. Nauchn. Semin. POMI,191, 101–113 (1991). [5] V. V. Ishkhanov and B. B. Lur’e, ”On the embedding problem with non-Abelian kernel of orderp 4. III,”Zap. Nauchn. Semin. POMI,198, 20–27 (1991). [6] V. V. Ishkhanov and B. B. Lur’e, ”On the embedding problem with non-Abelian kernel of orderp 4. IV,”Zap. Nauchn. Semin. POMI,211, 120–126 (1994). · Zbl 0871.12004 [7] V. V. Ishkhanov and B. B. Lur’e, ”On the embedding problem with non-Abelian kernel of orderp 4. V,”Zap. Nauchn. Semin. POMI,211, 127–132 (1994). · Zbl 0871.12005 [8] V. V. Ishkhanov, B. B. Lur’e, and D. K. Faddeev,The Embedding Problem in Galois Theory [in Russian], Moscow (1990). · Zbl 0727.12006 [9] J. Neukirch, ”On solvable number fields,”Invent. Math.,53, 135–164 (1979). · Zbl 0447.12008 · doi:10.1007/BF01390030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.