Central \(p\)-extensions of \((p, p, \dots, p)\)-type Galois groups (Q1125891)

Let \(p\) be a prime and \(K\) a field of characteristic not \(p\) and containing the \(p\)th roots of unity. This paper is concerned with obstructions to embedding problems for central \(p\)-extensions of a given Galois extension of \(K\), principally those which are elementary \(p\)-abelian, and with explicitly constructing such extensions. This work generalizes results of \textit{J.-P. Serre} [Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014)] and \textit{A. Fröhlich} [J. Reine Angew. Math. 360, 84-123 (1985; Zbl 0556.12005)] on obtaining obstructions to embedding problems, and results of \textit{T. Crespo} [J. Reine Angew. Math. 409, 180-189 (1990; Zbl 0696.12020); Proc. Am. Math. Soc. 112, 637-639 (1991; Zbl 0745.11051)]. Explicit formulas are given for the construction of fields \(N/K\) with Galois group a central \(p\)-extension of an elementary abelian \(p\)-group. Such constructions have also been considered by \textit{R. Massy} [J. Algebra 109, 508-535 (1987; Zbl 0625.12011)]. The methods used here, however, improve upon the constructions given by Massy in that they do not require the solution of a nontrivial system of linear equations in the field extension.