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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References:

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