## On $$p$$-groups as Galois groups.(English)Zbl 0707.12001

A group $$G$$ is said to be “realizable” over a field $$K$$ if there exists a Galois extension $$N/K$$, the Galois group of which is isomorphic to $$G$$; the field $$N$$ is then called a “$$G$$-extension” of $$K$$. Let $$p$$ be a prime number, and denote $$D_1,D_2$$ the two non-abelian groups of order $$p^3$$. The paper begins with a study of the “automatic realizations” of the pair $$D_1,D_2$$ in the sense of C. U. Jensen [Théorie des nombres, C. R. Conf. Int., Québec 1987, 441–458 (1989; Zbl 0696.12019)]. In the second section, the author solves the “descent problem” from a $$D_i$$-extension $$(i=1,2)$$ of $$K(\zeta)$$ to one of $$K$$, where $$K$$ is any field of characteristic different from $$p$$ and $$\zeta$$ a primitive $$p$$-th root of unity. She proves the new and interesting result that we explain now.
Let $$L/K$$ be a Galois extension with $$\Gamma = \mathrm{Gal}(L/K)$$ of type $$(p,p)$$ and let $$A$$ be a group of order $$p$$. The Galois group $$\Gamma'$$ of the translated extension $$L':=L(\zeta)/K':=K(\zeta)$$ is canonically isomorphic to $$\Gamma$$. Consider an embedding problem $$(L/K,\varepsilon)$$ with $$\varepsilon \in H^2(\Gamma,A)-\{0\}$$ and let $$N'/K'$$ be a solution of the problem $$(L'/K',\varepsilon')$$ where $$\varepsilon'\in H^2(\Gamma',A)$$ is the translated class of $$\varepsilon$$. By Kummer theory, there exists $$x\in L'^{\times}-L^{'\times p}$$ such that $$N'=L'(x^{1/p})$$. The key of the descent is the map $$\gamma:=n\sum_{\rho \in H} i(\rho)\rho^{-1}$$ where $$\rho (\zeta)=\zeta^{i(\rho)}$$ for all $$\rho \in H:= \mathrm{Gal}(K'/K)$$, and $$n | H| \equiv 1 \bmod p$$. Indeed, for $$N:=L'((\gamma x)^{1/p})$$, we have the
Theorem: (i) $$N$$ is Galois over $$K'$$ with $$\mathrm{Gal}(N/K')\overset{\sim}{\longrightarrow} \mathrm{Gal}(N'/K')$$; (ii) $$N$$ is Galois over $$K$$ with $$\mathrm{Gal}(N/K)\overset{\sim}{\longrightarrow} H\times \mathrm{Gal}(N'/K')$$ (direct product).
Corollary: There exists a $$D_i$$-extension $$N_0$$ of $$K$$ such that $$N=N_0(\zeta)$$ $$(i=1,2)$$.
In the last section, the author shows that there is a fixed numerical relation between the number of $$D_1$$-extensions and the number of $$D_2$$-extensions of $$K$$, provided one of these numbers is finite. The proofs use the theory of groups and reviewer’s results in [J. Algebra 109, 508–535 (1987; Zbl 0625.12011)]. Note that the theorem cited above is completely explicit since there exist formulae to get all the elements $$x$$ (the reviewer, loc. cit.).

### MSC:

 12F12 Inverse Galois theory

### Citations:

Zbl 0696.12019; Zbl 0625.12011
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