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Embedding problems with kernel of order two. (English) Zbl 0683.12012

Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 27-34 (1988).
[For the entire collection see Zbl 0653.00005.]
This is a report on some recent work of T. Crespo concerning a remark on embedding problems made by J.-P. Serre [Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014)]. Namely, let G be any subgroup of the alternating group \(A_ n\), and \(\tilde G\) be the pre-image of G in the double cover \(\tilde A_ n\) of \(A_ n\). Moreover, let \(E_ 0/K\) be a separable extension of degree \( n(\geq 4)\) over any field of characteristic \(\neq 2\) whose Galois closure E/K has a Galois group G(E/K) isomorphic to G.
Then, T. Crespo obtained the following result: the embedding problem \(\tilde G\to G(E/K)\) is solvable if and only if the Witt invariant of the trace form attached to the field \(E_ 0\) is trivial; in this case he also gave effectively the general solution to the problem. This is a generalization of Witt’s criterion to much more general situation given by Serre [cf. E. Witt, J. Reine Angew. Math. 174, 237-245 (1936; Zbl 0013.19601)].
Reviewer: H.Yokoi

MSC:

12F10 Separable extensions, Galois theory
11R32 Galois theory