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Construction de p-extensions galoisiennes d’un corps de caractéristique différente de p. (Construction of Galois p-extensions of fields with characteristic different from p). (French) Zbl 0625.12011

The following type of embedding problem is explicitly solved: given an extension of fields E/K with Galois group \(G=Gal(E/K)\) an elementary abelian p-group and a group extension \[ 1 \to {\mathbb{F}}_ p \to U \to G \to 1 \tag{*} \] where \({\mathbb{F}}_ p\) denotes a cyclic group of order p, do there exist extensions of fields N/E with Gal(N/E)\(\simeq {\mathbb{F}}_ p\) and Gal(N/K)\(\simeq U\), and, if the answer is affirmative, construct these fields N. It is assumed that the characteristic of K is not p and that K contains a primitive p-th root of unity \(\zeta_ p.\)
General results on embedding problems and on the Brauer group would only give the following information. A certain canonical homomorphism \(\Phi_{E/K}\) from the second cohomology group \(H^ 2(G,{\mathbb{F}}_ p)\) to the p-torsion subgroup of the Brauer group of K, \(Br(K)_ p\), has the property that the embedding problem has solutions iff the class \(\epsilon\) of (*) in \(H^ 2(G, {\mathbb{F}}_ p)\) lies in the kernel of \(\Phi_{E/K}\); moreover it is known that each element in \(Br(K)_ p\) can be written as a sum of elements of type (a,b) with \(a,b\in K^{\times 2}\), where (, ) denotes the usual symbol in \(Br(K)_ p.\)
Now in this paper the author gives the following more precise results: for all \(a,b\in K^{\times}\cap E^{\times p}\) elements \((a)_ E\) and \((a,b)_ E\) in \(H^ 2(G, {\mathbb{F}}_ p)\) are defined, it is verified that \(\Phi_{E/K}((a,b)_ E)=(a,b)\) and \(\Phi_{E/K}((a)_ E)=(a,\zeta_ p)\), and it is shown that and how \(\epsilon\) can be written in a canonical way as a sum of elements \((a,b)_ E\) and \((a)_ E\) (here three cases have to be distinguished and if \(p=2\) the classification of non-degenerate quadratic forms over \({\mathbb{F}}_ 2\) is used).
Furthermore the condition \(\Phi_{E/K}(\epsilon)=0\) is then rewritten in terms of a system of ‘norm equations’ and it is shown how each solution of this system gives a solution N of the embedding problem.
Reviewer: J.Brinkhuis

MSC:

12F10 Separable extensions, Galois theory
11R32 Galois theory
12G05 Galois cohomology
11R34 Galois cohomology
11E04 Quadratic forms over general fields
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